Categorification of skew-symmetrizable cluster algebras

CA TEGORIFICA TION OF SKEW-SYMMETRIZABLE CLUSTER ALGEBRAS LA URENT DEMONET Abstra t. W e prop ose a new framew ork for ategorifying sk ew-symmetrizable luster alge- bras. Starting from an exat stably 2 -Calabi-Y au ategory C endo w ed with the ation of a nite group Γ , w e onstrut a Γ -equiv arian t m utation on the set of maximal rigid Γ -in v arian t ob jets of C . Using an appropriate luster  harater, w e an then atta h to these data an expliit sk ew-symmetrizable luster algebra. As an appliation w e pro v e the linear indep endene of the luster monomials in this setting. Finally , w e illustrate our onstrution with examples asso i- ated with partial ag v arieties and unip oten t subgroups of Ka-Mo o dy groups, generalizing to the non simply-laed ase sev eral results of Geiÿ-Leler-S hrö er. Contents 1. In tro dution 1 1.1. Cluster algebras 1 1.2. A tions of groups on ategories 2 1.3. Categoriation of sk ew-symmetrizable luster algebras 4 1.4. Appliations 5 2. Equiv arian t ategories 5 2.1. Denitions and rst prop erties 5 2.2. A mo d k r Γ s -mo dule struture on C Γ 10 2.3. A k r Γ s -linear struture on the equiv arian t ategory 11 2.4. The funtors r Γ s and F 13 2.5. Appro ximations 17 2.6. A tion on an exat ategory 18 2.7. A tion on a F rob enius stably 2 -Calabi-Y au ategory 20 2.8. Computation of p k Q q Γ and p Λ Q q Γ 21 3. Categoriation of sk ew-symmetrizable luster algebras 22 3.1. Mutation of maximal Γ -stable rigid sub ategories 23 3.2. Rigid quasi-appro ximations 28 3.3. Endomorphisms 30 3.4. Ex hange matries 36 3.5. Cluster  haraters 38 3.6. Linear indep endene of luster monomials 41 4. Appliations 44 4.1. Reminder ab out ro ot systems and en v eloping algebras 44 4.2. Sub I J and partial ag v arieties 45 4.3. Categories C M and unip oten t groups 56 A  kno wledgmen ts 62 Referenes 62 1. Intr odution 1.1. Cluster algebras. In 2001, F omin and Zelevinsky in tro dued a new lass of algebras alled luster algebras [FZ1 ℄, [FZ2 ℄ motiv ated b y anonial bases and total p ositivit y [Lus1 ℄, [Lus3℄. By onstrution, a luster algebra is a omm utativ e ring endo w ed with distinguished generators 1 ( luster variables ) group ed in subsets of the same ardinalit y ( lusters ). The lusters are not disjoin t. On the on trary , ea h luster A has neigh b ours obtained b y replaing ea h of its v ariables x i b y a new v ariable x 1 i . The new luster µ i p A q  A zt x i u Y t x 1 i u is alled the m utation of A in the diretion x i . Moreo v er, the m utations are alw a ys of the form x i x 1 i  M i  M 1 i , where M i and M 1 i are monomials in the v ariables of A other than x i . The axioms imply strong ompatibilit y relations b et w een the monomials of the ex hange relations. In partiular, a luster algebra is fully determined b y a se e d , that is, a single luster and its ex hange relations with all its neigh b ours. In pratie, one usually denes a luster algebra b y giving an initial seed. By iterating the ex hange relations, one an express ev ery luster v ariable in terms of the v ariables of the initial seed. Berenstein, F omin and Zelevinsky ha v e sho wn that the o ordinate rings of man y algebrai v arieties atta hed to omplex semi-simple Lie groups w ere endo w ed with the struture of a luster algebra [BFZ℄. Other examples ha v e b een giv en b y Geiÿ, Leler and S hrö er [ GLS6 ℄, [GLS1 ℄. Sine their emergene, luster algebras ha v e aroused a lot of in terest, oming in partiular from their links with man y other sub jets: om binatoris (see for instane [CFZ℄, [FST ℄), P oisson geometry [GSV1℄, [GSV2℄, in tegrable systems [FZ3℄, T ei hm üller spaes [F G℄, and, last but not least, represen tations of nite-dimensional algebras. Unfortunately , b eause of the indutiv e desription of luster algebras, man y prop erties of the luster v ariables whi h migh t seem elemen tary are in fat v ery hard to pro v e. F or instane: Conjeture 1.1 (F omin-Zelevinsky) . Cluster monomials (that is pr o duts of luster variables of a single luster) ar e line arly indep endent. In seminal artiles, Marsh, Reinek e, Zelevinsky [MRZ℄, Buan, Marsh, Reinek e, Reiten, T o doro v [BMR  ℄ and Caldero, Chap oton [CC ℄ ha v e sho wn that the imp ortan t lass of ayli luster al- gebras ould b e mo delled with ategories onstruted from represen tations of quiv ers. This giv es in partiular a global (i.e. non indutiv e) understanding of these algebras, and giv es new to ols for studying them. F or example, this allo w ed F u and Keller [FK℄ to pro v e the previous onjeture for a family of luster algebras on taining ayli luster algebras. A t the same time, Geiÿ, Leler and S hrö er ha v e studied luster algebras asso iated with Lie groups of t yp e A , D , E , and ha v e mo delled them b y ategories of mo dules o v er Gelfand- P onomarev prepro jetiv e algebras of the same t yp e. They ha v e sho wn that luster monomials form a subset of the dual semi-anonial basis [GLS4 ℄ in tro dued b y Lusztig [Lus4 ℄, pro ving the ab o v e onjeture in this other on text. More reen tly , Derksen, W eyman and Zelevinsky [D WZ2 ℄, [D WZ1 ℄ ha v e obtained a far- rea hing generalization of [MRZ ℄, whi h also on tains all the ab o v e examples. They ha v e sho wn that one an on trol F -p olynomials and g -v etors of ev ery luster algebra whose initial seed is eno ded b y a sk ew-symmetri matrix, using represen tations of quiv ers with p oten tials. This enabled them to pro v e the linear indep endene onjeture, as w ell as man y other onjetures on F -p olynomials and g -v etors form ulated in [FZ4℄. But the theory of F omin and Zelevinsky inludes more general seeds giv en b y sk ew-symmetri- zable matries (i.e. pro duts of a sk ew-symmetri matrix b y a diagonal matrix). F or example, luster algebras asso iated to Lie groups of t yp e B , C , F , G are only sk ew-symmetrizable. The aim of this artile is to extend the results of Geiÿ, Leler, S hrö er and F u, Keller to the sk ew-symmetrizable ase. 1.2. A tions of groups on ategories. It is helpful to view a sk ew-symmetri matrix  M  r r m ij s P M n p Z q as an orien ted graph Q (i.e. a quiv er) with v ertex set Q 0  t 1 , 2 , . . . , n u and r m ij arro ws from i to j if r m ij ¡ 0 (resp. from j to i if r m ij  0 ). If a group Γ ats on Q , one an asso iate with it a new matrix M indexed b y the orbit set Q 0 { Γ , b y dening m ij as the n um b er of arro ws of Q b et w een a xed v ertex j of the orbit j and an y v ertex of the orbit i (oun ted 2 p ositiv ely if the arro ws go from i to j and negativ ely if they go from j to i ). It is easy to see that M is sk ew-symmetrizable. F or example, if  M      0 0 0 1 0 0 0 1 0 0 0 1  1  1  1 0  Æ Æ  then the quiv er Q is of t yp e D 4 1   ? ? ? ? ? ? ? ? 2 / / 4 3 . ? ?         There is an arro w from 1 to the orbit of 4 hene m 41   1 and there are three arro ws from the orbit of 1 to 4 hene m 14  3 . Th us w e obtain the matrix M   0 3  1 0  of t yp e G 2 1 < 4 . Hene the ation of a group Γ on a sk ew-symmetri matrix  M giv es rise to a sk ew-symme- trizable matrix M . If  M is the initial seed of a luster algebra r A ategoried as b efore b y a ategory C , it is natural to try to ategorify the luster algebra A with seed M b y a ategory C 1 onstruted from C and the group Γ . This leads to study k -additiv e ategories C on whi h a group Γ ats b y auto-equiv alenes. In this situation, one an form a ategory C Γ whose ob jets are pairs p X, p ψ g q g P Γ q onsisting of an ob jet X of C isomorphially in v arian t under Γ , together with a family of isomorphisms ψ g from X to ea h of its images b y the elemen ts g of Γ . One also requires that the ψ g satisfy natural ompatibilit y onditions. The ategory C Γ will b e alled the Γ -equiv arian t ategory . One then sho ws useful results of transfer. F or example:  if C is ab elian, then C Γ is ab elian;  if C is exat and if for all g P Γ the auto-equiv alene of C asso iated to g is exat, then C Γ is exat;  if H is a normal subgroup of Γ then Γ { H ats on C H and one has an equiv alene of ategories p C H qp Γ { H q  C Γ . W e also pro v e that C Γ an b e endo w ed with a natural ation of the ategory mo d k r Γ s of represen tations of Γ o v er k . The ategories C used b y Geiÿ, Leler, S hrö er and F u, Keller for mo delling luster algebras alw a ys ha v e the follo wing essen tial prop erties. They are F rob enius ategories (i.e. exat ate- gories with enough injetiv es and pro jetiv es, and the injetiv es and pro jetiv es are the same), and they satisfy Ext 1 C p X, Y q  Ext 1 C p Y , X q  , funtorially in X and Y . T o summarize, su h a ategory C is said to b e 2 -Calabi-Y au. In this framew ork, the notion of luster-tilting ob jet in tro dued b y [Iy a2℄ is v ery useful. An ob jet X is luster-tilting if it is rigid, that is, if Ext 1 C p X, X q  0 and if ev ery ob jet Y satisfying Ext 1 C p X, Y q  0 is in the additiv e en v elop e of X . If C ategories a sk ew-symmetri luster algebra r A , the luster-tilting ob jets mo del the lusters of r A , and their indeomp osable diret summands orresp ond to luster v ariables. Our prinipal transfer result sho ws that if C is 2 -Calabi-Y au, then C Γ is also 2 -Calabi-Y au. Moreo v er, the t w o natural adjoin t funtors linking C and C Γ indue reipro al bijetions b et w een 3 isomorphism lasses of Γ -stable luster-tilting ob jets of C and isomorphism lasses of mo d k r Γ s - stable luster-tilting ob jets of C Γ . In order to apply these general results to the examples studied b y Geiÿ, Leler and S hrö er, w e ha v e omputed expliitly the ategory C Γ in sev eral ases [ Dem1 ℄. This inludes in partiular the ase when C is the mo dule ategory of a prepro jetiv e algebra. 1.3. Categoriation of sk ew-symmetrizable luster algebras. Consider a 2 -Calabi-Y au ategory C on whi h ats a nite group Γ . F or ompleting the ategoriation, one needs to dev elop a theory of m utations of mo d k r Γ s -stable luster-tilting ob jets of C Γ , or, equiv alen tly , of Γ -stable luster-tilting ob jets of C . In the sk ew-symmetri ase (whi h an b e seen as the ase where Γ is trivial), it is kno wn that su h a theory is p ossible as so on as there exists a luster- tilting ob jet whose asso iated quiv er has neither lo ops nor 2 -yles. W e in tro due in the general ase the onepts of mo d k r Γ s -lo ops and of mo d k r Γ s - 2 -yles for a mo d k r Γ s -stable luster- tilting ob jet. One then sho ws that if C Γ admits a mo d k r Γ s -stable luster-tilting ob jet ha ving neither mo d k r Γ s -lo ops nor mo d k r Γ s - 2 -yles, all mo d k r Γ s -stable luster-tilting ob jets also ha v e this prop ert y . Under this h yp othesis, one an dene a m utation op eration. More preisely , if T is mo d k r Γ s -stable luster-tilting and if X is the mo d k r Γ s -orbit of an indeomp osable non pro jetiv e diret summand X of T , one onstruts another mo d k r Γ s -stable luster-tilting ob jet T 1 obtained b y replaing X b y the mo d k r Γ s -orbit Y of another indeomp osable ob jet Y . One denotes µ X p T q  T 1 . One an also asso iate to T a sk ew-symmetrizable matrix B p T q whose ro ws are indexed b y the mo d k r Γ s -orbits X of indeomp osable summands of T and the olumns b y the mo d k r Γ s -orbits X of indeomp osable non pro jetiv e fators of T . The o eien ts b X Y are the n um b ers of arro ws in the Gabriel quiv er of End C p T q b et w een a xed indeomp osable ob jet Y of Y and an y indeomp osable ob jet X of X (the arro ws from X to Y b eing oun ted p ositiv ely and the arro ws from Y to X b eing oun ted negativ ely). W e then sho w (see theorem 3.42 ): Theorem A. The mutation of Γ -stable luster-tilting obje ts of C agr e es with the mutation dene d  ombinatorial ly by F omin and Zelevinsky for skew-symmetrizable matri es. That is, B p µ X p T qq  µ X p B p T qq , wher e, in the right-hand side, by abuse of notation, µ X is the matrix mutation of F omin and Zelevinsky. Via the ab o v e-men tioned bijetion b et w een mo d k r Γ s -stable luster-tilting ob jets of C Γ and Γ -stable luster-tilting ob jets of C , one an asso iate to ea h Γ -stable luster-tilting ob jet T of C a matrix whi h will b e also denoted b y B p T q . Finally , in order to atta h to C and Γ a luster algebra, w e in tro due a notion of Γ -equiv arian t luster  harater. In the sk ew-symmetri ase, aording to the w ork of Caldero-Chap oton [ CC℄, of Caldero-Keller [CK2 ℄, [CK1℄, of P alu [P al ℄, of Deh y-Keller [DK℄ and of F u-Keller [FK ℄ one an assign to ev ery ob jet X of C a Lauren t p olynomial in the luster v ariables of an initial seed of the luster algebra A ategoried b y C . If this initial seed is Γ -stable, one an iden tify the luster v ariables whi h b elong to the same Γ -orbit. This sp eialization of the Lauren t p olynomial asso iated to X only dep ends on the Γ -orbit X of X , and is denoted b y P X . One dedues from the previous onstrution that if T is a Γ -stable luster-tilting ob jet in C , and if A is the luster algebra whose initial seed has sk ew-symmetrizable matrix B p T q , then the luster v ariables of A are of the form P X where X is the orbit of an indeomp osable summand of a Γ -stable luster- tilting ob jet of C . In this situation, w e sa y that the pair p C , Γ q is a ategoriation of A . One an then generalize the result of F u and Keller (see orollary 3.61 ): Theorem B. L et B b e an m  n matrix with skew-symmetrizable prinip al p art. If B has ful l r ank and if the luster algebr a A p B q has a  ate gori ation p C , Γ q , then the luster monomials ar e line arly indep endent. As a result, w e obtain a pro of of onjeture 1.1 for a large family of sk ew-symmetrizable luster algebras. 4 1.4. Appliations. Finally , w e giv e new families of examples of ategoriation of luster al- gebras. Let G b e a semi-simple onneted and simply-onneted Lie group of simply-laed Dynkin diagram ∆ , and let Λ b e the asso iated prepro jetiv e algebra. Geiÿ, Leler and S hrö er ha v e sho wn that the sub ategories Sub I J of mo d Λ indue luster strutures on the m ulti- homogeneous o ordinate rings of partial ag v arieties asso iated to G [GLS6 ℄. Our w ork allo ws to extend this result to the ase where G orresp onds to a non simply-laed Dynkin diagram. In partiular, one obtains a pro of of the onjeture 1.1 for these luster algebras, and one an omplete the lassiation of partial ag v arieties whose luster struture is of nite t yp e (i.e. admit a nite n um b er of lusters). In partiular, this pro v es the onjeture of [GLS6 , 14℄. Let G b e a Ka-Mo o dy group of symmetri Cartan matrix, and let Λ b e the asso iated prepro jetiv e algebra. Geiÿ, Leler and S hrö er ha v e in tro dued ertain sub ategories C M of mo d Λ and sho wn that they indue luster strutures on the o ordinate ring of some unip oten t subgroups and unip oten t ells of G [GLS1 ℄ (see also [BIRS℄ whi h giv es a dieren t denition of similar sub ategories). Our w ork allo ws to extend these results to the Ka-Mo o dy groups G with symmetrizable Cartan matries. In partiular, one obtains for all these examples a pro of of the onjeture 1.1 . As a partiular ase of this onstrution, w e get (see theorem 4.47 ): Theorem C. F or every ayli luster algebr a without  o eient A , ther e is a  ate gory C and a nite gr oup Γ ating on C whih  ate gorify A up to sp e ialization of  o eients to 1 . This holds in p artiular for luster algebr as of nite typ e. Note that the w orks of Geiÿ, Leler and S hrö er use as a ruial fat the existene of the dual semianonial basis onstruted b y Lusztig for the o ordinate ring of a maximal unip oten t subgroup of G . But, when G is not of simply-laed t yp e, there is no a v ailable onstrution of semianonial bases. Our result an b e in terpreted as giving a part of the dual semianonial basis in the non simply-laed ase, namely the set of luster monomials. 2. Equiv ariant a tegories F or referenes ab out monoidal ategories and mo dule ategories o v er a monoidal ategory , see for example [BK ℄, [CP ℄, [Kas ℄ and [Ost℄. 2.1. Denitions and rst prop erties. Let k b e a eld, C a k -ategory , Hom -nite and Krull- S hmidt (whi h means that the endomorphism rings of indeomp osable ob jets are lo al, or equiv alen tly , that ev ery idemp oten t splits). Let Γ b e a nite group whose ardinalit y is not divisible b y the  harateristi of k . Let Γ  mo d k p Γ q b e the monoidal ategory of k p Γ q -mo dules, where k p Γ q is the Hopf algebra of k -v alued funtions on the group Γ . Remark that the simple ob jets in Γ are the one-dimensional k p Γ q -mo dules giv en b y ev aluation maps at ea h elemen t g of Γ . If g P Γ , the orresp onding simple ob jet in Γ will b e denoted b y g . With this notation, it is easy to  he k that the monoidal struture is simply g b h  gh , where for g , h P Γ , w e denote b y gh the simple k p Γ q -mo dule orresp onding to g h P Γ . Denition 2.1. An ation of Γ on C is a struture of Γ -mo dule ategory on C . R emark 2.2 . If one onsiders, as in [RR , p. 254℄, a group morphism ρ from Γ to the group of autofuntors of C , one obtains a strit Γ -mo dule struture b y setting g b   ρ p g q . W e no w in tro due a ategory of Γ -in v arian t ob jets of C . The naiv e idea of onsidering the full sub ategory of C of in v arian t ob jets do es not w ork b eause almost none of the desired prop erties are preserv ed. Denition 2.3. Let C b e endo w ed with an ation of Γ . The Γ -e quivariant  ate gory of C is the ategory whose ob jets are pairs p X, ψ q , where X P C , and ψ  p ψ g q g P Γ is a family of 5 isomorphisms ψ g : g b X Ñ X su h that, for ev ery g , h P Γ , the follo wing diagram omm utes: g b p h b X q Id g b ψ h / / g b X ψ g   gh b X α O O ψ gh / / X. Here, α denotes the Γ -mo dule strutural isomorphism. W e also assume that ψ e : 1 b X Ñ X is the strutural isomorphism of the Γ -mo dule ategory whenev er e is the neutral elemen t of Γ . The morphisms from an ob jet p X, ψ q to an ob jet p Y , χ q are the morphisms f from X to Y su h that for ev ery g P Γ , the follo wing diagram omm utes: g b X Id g b f   ψ g / / X f   g b Y χ g / / Y . Notation 2.4.  In the sequel, in partiular in diagrams, w e will denote b y ψ ev ery arro w of the form Id b ψ g .  Ev ery Γ -mo dule strutural isomorphism will b e denoted b y α in diagrams.  The Γ -equiv arian t ategory of C will b e denoted b y C Γ . R emark 2.5 . This ategory is equiv alen t to the sk ew group ategory onsidered b y Reiten and Riedtmann in [RR , p. 254℄. W e ha v e found our denition easier to handle b eause it do es not require to use a Karoubi en v elop e. Moreo v er, it p ermits to deal with non-strit ations, whi h is more pratial in ertain ases. F or more details ab out this problem, in partiular for the pro of of the equiv alene, see [Dem3 ℄. Examples 2.6. (i) If C  mo d k is the ategory of nite dimensional k -v etor spaes, and the ation of Γ on C is trivial, then C Γ  mo d k r Γ s . (ii) If A is a k -algebra and Γ ats on A , Γ ats naturally on mo d A . Then, one has p mo d A q Γ  mo d p AG q where AG is the sk ew group algebra dened in [RR ℄ (see also setion 2.8 ). (iii) If Γ is a yli group, the ategory C Γ is the same as the one onsidered in [ Lus2 ,  hapter 11℄. The follo wing prop osition is an easy generalization of a prop osition of Gabriel [ Gab, p. 9495℄: Prop osition 2.7. If Γ  x g 0 y is yli of or der n P N , and if every element of k has an n -th r o ot in k , then every X P C suh that X  g 0 b X has a lift p X, ψ q in C Γ . Note that prop osition 2.7 do es not generalize to a non yli group (see e.g. [ Dem3, ex. 2.1.19℄). Lemma 2.8. (i) The  ate gory C Γ is k -additive, Hom -nite and Krul l-Shmidt. (ii) If C is ab elian, then C Γ is also ab elian. (iii) If C is exat and if for al l g P Γ , the funtor g b  : C Ñ C is exat (one wil l say that the ation is exat), then C Γ is exat with admissible short exat se quen es of the form 0 Ñ p X, ψ q f Ý Ñ p X 1 , ψ 1 q g Ý Ñ p X 2 , ψ 2 q Ñ 0 suh that 0 Ñ X f Ý Ñ X 1 g Ý Ñ X 2 Ñ 0 is an admissible short exat se quen e in C . Pr o of. All these p oin ts are lear from the funtorialit y of the v arious onstrutions (k ernel, splitting, . . . ).  Denition 2.9. Let n P N . An n -asso iativity is a funtor from Γ n  C to C built from the bifuntor b and the ob jet 1 P Γ . F or instane, p b q b p 1 b q and  b p b q are 2 -asso iativities. An asso iativity is an n -asso iativit y for some n . 6 Let H b e a subgroup of Γ and p X, ψ q P C H . Let g 1 , g 2 , . . . , g n , g 1 1 , g 1 2 , . . . , g 1 m P Γ . Let A 1 b e an n -asso iativit y and A 2 an m -asso iativit y . A ψ -strutur al isomorphism from A 1 p g 1 , . . . , g n , X q to A 2 p g 1 1 , . . . , g 1 m , X q is an y isomorphism omp osed of strutural isomorphisms of the Γ -mo dule struture, and of isomorphisms of the form Φ p ψ h q or Φ p ψ  1 h q where h P H and Φ is a funtor onstruted from an asso iativit y and ob jets of Γ . Lemma 2.10 (oherene) . L et H b e a sub gr oup of Γ . L et p X, ψ q P C H . L et g 1 , g 2 , . . . , g n , g 1 1 , g 1 2 , . . . , g 1 m P Γ . L et also A 1 b e an n -asso iativity and A 2 b e an m -asso iativity. Then al l ψ -strutur al isomor- phisms fr om A 1 p g 1 , g 2 , . . . , g n , X q to A 2 p g 1 1 , g 1 2 , . . . , g 1 m , X q ar e e qual. Pr o of. By in v ertibilit y of the ψ -strutural isomorphisms, one an supp ose that A 1  A 2 and that one of the t w o ψ -strutural isomorphisms is the iden tit y . Let f b e a ψ -strutural isomorphism from A 1 p g 1 , g 2 , . . . , g n , X q to itself. By using omm utativ e diagrams built from diagrams of the form X α   h b X α   ψ o o h b p h  1 b X q ψ   h b p h  1 b p h b X qq α   ψ o o h b X h b p 1 b X q ψ q q α m m with h P H , one an supp ose that f on tains only p ositiv e p o w ers of ψ . By using omm utativ e diagrams of the form Φ 1 p X q α / / Φ 2 p h b X q ψ / / Φ 2 p X q Φ 1 p h 1 b X q ψ O O α / / Φ 2 p h b p h 1 b X qq ψ O O α / / Φ 2 p hh 1 b X q ψ O O where h, h 1 P H and Φ 1 and Φ 2 are funtors made from an asso iativit y and ob jets of Γ , one an supp ose that f is of the form α 1 Φ p ψ h q α 2 where α 1 and α 2 are strutural morphisms of C and Φ is made from an asso iativit y and ob jets of Γ and h P H . Then, as α 1 Φ p ψ h q α 2 go es from A 1 p g 1 , g 2 , . . . , g n , X q in to itself, h is the neutral elemen t of H and as ψ e is strutural, f is a strutural morphism of the Γ -mo dule struture whi h implies the result, b y the MaLane oherene lemma (see [Ma ℄).  Prop osition 2.11. L et H b e a normal sub gr oup of Γ . The ation of Γ on C indu es an ation of H on C . Then: (i) The ation of Γ on C extends to an ation of Γ on C H . (ii) F or every h P H , ther e is an isomorphism of funtors fr om h b  to Id C H . (iii) The ation of Γ on C H indu es an ation of Γ { H on C H . (iv) Ther e is an e quivalen e of  ate gories b etwe en p C H qp Γ { H q and C Γ . Pr o of. (i) Let p X, ψ q P C H . If g P Γ and h P H , dene p g b ψ q h su h that the follo wing diagram omm utes: g b X g b p g  1 hg b X q α v v l l l l l l l l l l l l l ψ o o h b p g b X q p g b ψ q h O O 7 F or g P Γ and h, h 1 P H , the follo wing diagram is omm utativ e from lemma 2.10 b eause all arro ws are ψ -strutural: h b p h 1 b p g b X qq h bp g b ψ q h 1 / / h b p g b X q p g b ψ q h   hh 1 b X α O O p g b ψ q hh 1 / / g b X whi h means that g b p X, ψ q  p g b X, g b ψ q P C H (it is easy to see that p g b ψ q e is strutural). The follo wing diagram omm utes for g , g 1 P Γ and h P H : h b p g b p g 1 b X qq p g bp g 1 b ψ qq h / / α   g b p g 1 b X q α   h b p gg 1 b X q p gg 1 b ψ q h / / gg 1 b X. Therefore the strutural isomorphisms of C are also strutural isomorphisms of C H whi h leads to the onlusion. (ii) F or h, h 1 P H , for the same reason, the follo wing diagram omm utes: h 1 b p h b X q p h b ψ q h 1   ψ / / h 1 b X ψ   h b X ψ / / X. Therefore ψ h is an isomorphism from h b X to X in the ategory C H . If p Y , χ q P C H and f : p X, ψ q Ñ p Y , χ q is a morphism, the follo wing diagram omm utes: h b X ψ / / Id h b f   X f   h b Y χ / / Y . If one denotes ϕ h ; Y ,χ  χ h for ev ery p Y , χ q P C H , then ϕ h is an isomorphism from the funtor h b  to the funtor Id C H . (iii) F or g P Γ , denote b y g its lass in Γ { H . Let Γ 0  Γ b e a set of represen tativ es of Γ { H on taining the neutral elemen t. F or ev ery g P Γ 0 , let g b   g b  . Let g , g 1 P Γ 0 . There exists a unique deomp osition g g 1  g 2 h with g 2 P Γ 0 and h P H . The isomorphism of funtors α , su h that the diagram gg 1 b  α   g 2 b  g 2 b p h b q ϕ h o o g b p g 1 b q g b p g 1 b q α / / g 2 h b  α O O omm utes, endo ws C H with an ation of Γ { H . Indeed, all axioms of a Γ { H -mo dule ategory are v eried b eause ev ery morphism onsidered in these axioms is ψ -strutural and therefore the equalities are true. 8 (iv) Let p X, ψ 1 , ψ 2 q P p C H qp Γ { H q . F or g P Γ , there exists g 0 P Γ 0 and h P H su h that g  g 0 h . Let ψ g b e su h that the follo wing diagram omm utes: g b X α / / ψ g   g 0 b p h b X q ψ 1 / / g 0 b X X g b X ψ 2 o o Let no w g 1 P Γ , g 1 0 P Γ 0 , h 1 P H , g 2 0 P Γ 0 and h 2 P H su h that g 1  g 1 0 h 1 and g g 1  g 2 0 h 2 . The follo wing diagram omm utes: g b p g 1 0 b p h 1 b X qq ψ 1 / / α   g b p g 1 0 b X q α   ψ 2 / / g b X α   g 0 b p h b p g 1 0 b X qq ψ 2 / / g 1 0 b ψ 1   g 0 b p h b X q ψ 1   g 0 b p h b p g 1 0 b p h 1 b X qqq ψ 1 4 4 h h h h h h h h h h h h h h h h h g 0 b p g 1 0 b X q ψ 2 / / g 0 b X ψ 2   g 2 0 b p h 2 b X q ψ 1 / / α O O g 2 0 b X ψ 2 / / α O O X (the t w o upp er squares b eause α is funtorial, the middle righ t square b eause ψ 2 is a morphism from g 1 0 b p X, ψ 1 q to p X, ψ 1 q , the lo w er righ t square b y denition of ψ 2 and the lo w er left p en tagon b eause it is formed b y ψ 1 -strutural morphisms). Keeping only the b order and omp osing with strutural morphisms on the left, one gets the follo wing omm utativ e diagram: g b p g 1 b X q ψ / / α   g b X ψ   gg 1 b X ψ / / X hene p X, ψ q P C Γ . Let Φ p X, ψ 1 , ψ 2 q  p X, ψ q . Let p Y , χ 1 , χ 2 q P p C H qp Γ { H q b e an- other ob jet and p Y , χ q  Φ p Y , χ 1 , χ 2 q . Let f P Hom p C H qp Γ { H q p X, ψ 1 , ψ 2 ; Y , χ 1 , χ 2 q . The follo wing diagram omm utes: g b X α / / f   ψ ' ' g 0 b p h b X q f   ψ 1 / / g 0 b X f   ψ 2 / / X f   g b Y α / / χ 7 7 g 0 b p h b Y q χ 1 / / g 0 b Y χ 2 / / Y and, as a onsequene, f P Hom C Γ pp X, ψ q , p Y , χ qq . By setting Φ p f q  f , Φ is a funtor. No w, if p X, ψ q P C Γ , for h P H , let ψ 1 h  ψ h and for g P Γ 0 , ψ 2 g 0  ψ g 0 . It is easy to  he k that p X, ψ 1 , ψ 2 q P p C H qp Γ { H q (ea h in v olv ed morphism is ψ -strutural) and that Φ p X, ψ 1 , ψ 2 q  p X, ψ q . Finally , Φ is essen tially surjetiv e. Moreo v er, it is easy to see that Φ is fully faithful. Hene Φ is an equiv alene of ategories.  9 2.2. A mo d k r Γ s -mo dule struture on C Γ . W e denote b y k r Γ s the group algebra of Γ . This is a Hopf algebra (dual to k p Γ q ), hene mo d k r Γ s is a monoidal ategory . An ob jet of mo d k r Γ s will b e denoted b y p V , r q , where V is a k -v etor spae, and r : Γ Ñ GL p V q a group homomorphism. Prop osition 2.12. The  ate gory C Γ is a mo d k r Γ s -mo dule  ate gory in a natur al way. Pr o of. Let mo d 0 k b e the full sub ategory of mo d k , whose ob jets are k n ( n P N ). Let Ψ b e the inlusion funtor. It is easy to extend Ψ to a monoidal equiv alene of ategories. Let Φ b e a monoidal quasi-in v erse of Ψ . One endo ws C with a struture of mo d k -mo dule ategory b y setting for ev ery V P mo d k and X P C , V b X  X dim p V q . If V , W P mo d k , X, Y P C , f P Hom k p V , W q and g P Hom C p X, Y q , one denes f b g : V b X  X dim p V q Ñ W b Y  Y dim p W q b y f b g  p Φ p f q ij g q 1 6 i 6 dim p W q , 1 6 j 6 dim p V q . F or ev ery V , W P mo d k and X P C , α V ,W,X : p V b W q b X Ñ V b p W b X q is dened b y α V ,W,X ; i,j, ℓ  " Id X if i  j  dim p V qp ℓ  1 q , 0 else , . where 1 6 i 6 dim p V b W q  dim p V q dim p W q , 1 6 j 6 dim p V q and 1 6 ℓ 6 d im p W q . It is no w easy to  he k that C is mo d 0 k -mo dule with these strutural isomorphisms. As Φ is a monoidal equiv alene, C is also mo d k -mo dule. One dedues that C Γ is mo d k r Γ s -mo dule. Indeed, one remarks that for ev ery g P Γ , V P mo d k and X P C , one has g b p V b X q  g b X dim V  p g b X q dim V  V b p g b X q . If p V , r q P mo d k r Γ s and p X, ψ q P C Γ , let p V , r q b p X, ψ q  p V b X, r b ψ q where, for g P Γ , p r b ψ q g : g b p V b X q  V b p g b X q Ñ V b X is dened b y p r b ψ q g  r g b ψ g . If one tak es also h P Γ , one gets the t w o omm utativ e diagrams V r h / / V r g   g b p h b X q Id g b ψ h / / g b X ψ g   V r gh / / V gh b X α O O ψ gh / / X and applying the bifuntor b : mo d k  C Ñ C yields the omm utativ e diagram g b p h b p V b X qq Id g bp r b ψ q h / / g b p V b X q p r b ψ q g   gh b p V b X q α O O p r b ψ q gh / / V b X hene p V b X, r b ψ q P C Γ . Moreo v er, if p V 1 , r 1 q P mo d k r Γ s , p Y , χ q P C Γ , f P Hom mod k r Γ s pp V , r q , p V 1 , r 1 qq and f 1 P Hom C Γ pp X, ψ q , p Y , χ qq , the t w o follo wing diagrams omm ute for ev ery g P Γ : V f   r g / / V f   g b X Id g b f 1   ψ g / / X f 1   V 1 r 1 g / / V 1 g b Y χ g / / Y whi h sho ws b y applying the bifuntor b : mo d k  C Ñ C that f b f 1 P Hom C Γ pp V , r q b p X, ψ q , p V 1 , r 1 q b p Y , χ qq . This nishes the pro of that C Γ is a mo d k r Γ s -mo dule ategory .  R emark 2.13 . The previous struture do es not dep end on the  hoie of Φ up to isomorphism. 10 2.3. A k r Γ s -linear struture on the equiv arian t ategory. The aim of this setion is to dene new morphisms spaes on C Γ whi h are k r Γ s -mo dules. These new strutures will b e written in b old fae. This giv es a new ategory losely related to C Γ . The main relationships b et w een the t w o ategories will b e outlined. Notation 2.14. One denotes b y F the forgetful funtor from C Γ to C . Reall this lassial lemma: Lemma 2.15. Ther e is an isomorphism of trifuntors fr om p mo d k r Γ sq 3 to mo d k : Hom mod k r Γ s p ? 1 b ?  2 , ? 3 q  Hom mod k r Γ s p ? 1 , ? 2 b ? 3 q . If r is a k r Γ s -mo dule, r  denotes its  ontr agr e dient, or dual r epr esentation. Here is an easy lemma (for a detailed pro of, see [Dem3, lemme 2.1.16℄) : Lemma 2.16. With the pr evious notations, ther e is an isomorphism of quadrifuntors Hom C p ? 1 b  1 , ? 2 b  2 q  ?  1 b ? 2 b Hom C p 1 ,  2 q wher e the  ar e variables of C and the ? ar e variables of mo d k . Let p X, ψ q , p Y , χ q P C Γ . As a k -v etor spae, let Hom C Γ pp X, ψ q , p Y , χ qq  Hom C p X, Y q . If g P Γ and f P Hom C Γ p X, Y q , dene g f P Hom C Γ p X, Y q b y the follo wing omm utativ e diagram: X g f / / Y g b X Id g b f / / ψ g O O g b Y χ g O O Th us, one will pro v e in prop osition 2.17 that Hom C Γ p X, Y q aquires the struture of a k r Γ s - mo dule. If Γ : p X 1 , ψ 1 q Ñ p X, ψ q , Γ 1 : p Y , χ q Ñ p Y 1 , χ 1 q are morphisms in C Γ , dene Hom C Γ p Γ , Γ 1 q  Hom C p F Γ , F Γ 1 q . Prop osition 2.17. Dene d in this way, Hom C Γ is a bifuntor fr om C Γ  C Γ to mo d k r Γ s  ontr avariant in the rst variable and  ovariant in the se  ond one whih satises: (i) for X, Y , Z P C Γ the  omp osition  : Hom C p F Y , F Z q b Hom C p F X , F Y q Ñ Hom C p F X , F Z q is a morphism of k r Γ s -mo dules  : Hom C Γ p Y , Z q b Hom C Γ p X, Y q Ñ Hom C Γ p X, Z q ; (ii) ther e is an isomorphism of quadrifuntors Hom C Γ p ? 1 b  1 , ? 2 b  2 q  ?  1 b ? 2 b Hom C Γ p 1 ,  2 q wher e the ? ar e variables in mo d k r Γ s and the  ar e variables in C Γ ; (iii) ther e is an isomorphism of quadrifuntors Hom C Γ p ? 1 b  1 , ? 2 b  2 q  Hom mod k r Γ s p ? 1 b ?  2 , Hom C Γ p 1 ,  2 qq wher e the ? ar e variables in mo d k r Γ s and the  ar e variables in C Γ . In p artiular it endows C Γ with the strutur e of a mo d k r Γ s -line ar  ate gory. 11 Pr o of. F or g , h P Γ , the follo wing diagram omm utes: X g p hf q / / Y g b X ψ O O Id g b hf / / g b Y χ O O gh b X ψ ? ?                    Id gh b f 2 2 g b p h b X q α o o ψ O O Id g bp Id h b f q / / g b p h b Y q χ O O α / / gh b Y χ ^ ^ > > > > > > > > > > > > > > > > > > > so g p hf q  p g h q f and Hom C Γ p X, Y q is a represen tation of Γ . If Γ : p X 1 , ψ 1 q Ñ p X, ψ q , Γ 1 : p Y , χ q Ñ p Y 1 , χ 1 q are morphisms in C Γ , and if f P Hom C p X, Y q , the follo wing diagram omm utes for ev ery g P Γ : X 1 g Hom C p Γ , Γ 1 qp f q / / Γ                   Y 1 g b X 1 ψ 1 O O Id g b Γ   Id g b Hom C p Γ , Γ 1 qp f q / / g b Y 1 χ 1 O O X g f 3 3 g b X ψ o o Id g b f / / g b Y Id g b Γ 1 O O χ / / Y Γ 1 Y Y 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 and nally , g Hom C p Γ , Γ 1 qp f q  Hom C p Γ , Γ 1 qp g f q . Hene Hom C p Γ , Γ 1 q is a morphism from Hom C Γ pp X, ψ q , p Y , χ qq to Hom C Γ pp X 1 , ψ 1 q , p Y 1 , χ 1 qq , morphism whi h will b e denoted Hom C Γ p Γ , Γ 1 q turning Hom C Γ in to a bifuntor. Let us no w pro v e the three additional prop erties: (i) is lear. (ii) It is enough to sho w that the isomorphism of quadrifuntors Γ dened in lemma 2.16 remains an isomorphism. Let g P Γ and p V , r q , p V 1 , r 1 q P mo d k r Γ s . By denition of a morphism of funtors, the lo w er square of the follo wing diagram omm utes: V  b V 1 b Hom C p X, Y q Γ / / Id b Id bp g bq   Hom C p V b X, V 1 b Y q g b   V  b V 1 b Hom C p g b X, g b Y q Γ / / t r  1 g b r 1 g b Hom C p ψ  1 g ,χ g q   Hom C p V b p g b X q , V 1 b p g b Y qq Hom C p r  1 g b ψ  1 g ,r 1 g b χ g q   V  b V 1 b Hom C p X, Y q Γ / / Hom C p V b X, V 1 b Y q and the upp er square omm utes b eause V b p g b X q  g b p V b X q . This pro v es that Γ V ,V 1 ,X,Y is a morphism of represen tations. (iii) F rom lemma 2.15 and (ii), one gets Hom mod k r Γ s p ? 1 b ?  2 , Hom C Γ p 1 ,  2 qq  Hom mod k r Γ s p 1 , ?  1 b ? 2 b Hom C Γ p 1 ,  2 qq  Hom mod k r Γ s p 1 , Hom C Γ p ? 1 b  1 , ? 2 b  2 qq hene it is enough to see that Hom mod k r Γ s p 1 , Hom C Γ p 1 ,  2 qq  Hom C Γ p 1 ,  2 q . This is lear. Indeed, it is suien t, if f P Hom mod k r Γ s p 1 , Hom C Γ pp X, ψ q , p Y , χ qqq , to asso- iate to it f p 1 q P £ g P Γ Hom C Γ pp X, ψ q , p Y , χ qq g  Hom C Γ pp X, ψ q , p Y , χ qq 12 the last equalit y b eing the denition of Hom C Γ pp X, ψ q , p Y , χ qq .  Corollary 2.18. Ther e is an isomorphism of trifuntors Hom C Γ p ?  b  , q  Hom C Γ p , ? b q wher e ? is a variable of mo d k r Γ s and the  ar e variables of C Γ . In p artiular, if r P mo d k r Γ s , the two funtors fr om C Γ to itself r b  and r  b  ar e adjoint. Pr o of. Using prop osition 2.17 , one gets Hom C Γ p ?  b  , q  Hom mod k r Γ s p ?  , Hom C Γ p , qq  Hom C Γ p , ? b q , whi h is the desired isomorphism.  2.4. The funtors r Γ s and F . Prop osition 2.19. F or X P C , dene r X   à g P Γ g  b X. F or g P Γ , denote by ψ g the unique strutur al isomorphism fr om g b r X to r X . Then p r X , ψ q P C Γ . Pr o of. F or g , h P Γ , the diagram g b p h b r X q Id g b ψ h / / g b r X ψ g   gh b r X α O O ψ gh / / r X omm utes b y uniit y of the strutural isomorphism from g b p h b r X q to r X (lemma 2.10 ).  Denition 2.20. The ob jet p r X , ψ q will b e denoted b y X r Γ s . Lemma 2.21. One  an extend r Γ s to a funtor fr om C to C Γ by setting f r Γ s   à g P Γ Id g  b f for every morphism f of C . Pr o of. It is enough to see that if f : X Ñ Y is a morphism in C , then f r Γ s   à g P Γ Id g  b f is a morphism from X r Γ s to Y r Γ s . Let h P Γ . The diagram h b  À g P Γ g  b X  Id bp Id b f q / / α   h b   À g P Γ g  b Y  α    À g P Γ g  b X Id b f / /  À g P Γ g  b Y omm utes b y funtorialit y of strutural morphisms.  Prop osition 2.22. (i) Ther e is an isomorphism of funtors fr om C Γ to itself: p F qr Γ s  k r Γ s b  wher e F : C Γ Ñ C is the for getful funtor and k r Γ s denotes the r e gular r epr esentation of Γ . (ii) The funtors F : C Γ Ñ C and r Γ s : C Ñ C Γ ar e adjoint. 13 Pr o of. (i) Let p X, ψ q P C . Let χ b e su h that  à g P Γ g  b X, χ   X r Γ s . F or g , h P Γ , the follo wing diagram omm utes: h b p g b X q α / / h b ψ g   hg b X ψ hg   h b X ψ h / / X. A dding up these diagrams for g P Γ , b y setting g 1  hg , the follo wing diagram omm utes: h b  À g g  b X  χ h / / h b À ψ g ' '  À g 1 g 1  b X À g p h b p g b X qq À α / / À h b ψ g   À g 1 p g 1 b X q À ψ g 1   À g p h b X q À ψ h / / À g 1 X h b  À g X  p k r Γ sb ψ q h / / À g 1 X the equalities onsisting of unique iden tiations through univ ersal prop erties. As a onsequene, À ψ g is an isomorphism from X r Γ s to k r Γ s b p X, ψ q . Moreo v er, if f : p X, ψ q Ñ p Y , ψ 1 q is a morphism, the follo wing diagram omm utes:  À g g  b X p À g qb f / /  À g g  b Y À g p g b X q À p g b f q / / À ψ g   À g p g b Y q À ψ 1 g   À g X k r Γ sb f  À f / / À g Y the omm utativit y of the lo w er square oming from the fat that f is a morphism in C Γ . Finally , À ψ g is a funtorial isomorphism. (ii) Let X P C and p Y , ψ q P C Γ . If f P Hom C p X, Y q , one denes ξ p f q : à g P Γ p g b X q   à g P Γ g  b X Ñ Y b y p ξ p f qq g  ψ g  p g b f q . F or g , h P Γ , the follo wing diagram omm utes: h b p g b X q h bp g b f q / / α   h b p g b Y q h b ψ g / / α   h b Y ψ h   hg b X hg b f / / hg b Y ψ hg / / Y . 14 By adding up these diagrams for g P Γ , one obtains the follo wing omm utativ e diagram: h b p p À g q b X q h b ξ p f q / / χ h   h b Y ψ h   p À g q b X ξ p f q / / Y whi h sho ws that ξ p f q is a morphism from X r Γ s to p Y , ψ q . W e no w  he k that ξ is funtorial. Let X 1 P C and p Y 1 , ψ 1 q P C Γ . Let η : X 1 Ñ X b e a morphism of C and θ : p Y , ψ q Ñ p Y 1 , ψ 1 q a morphism of C Γ . Then, for g P Γ , p θ  ξ p f q  η r Γ sq g  θ  ξ p f q g  p g b η q  θ  ψ g  p g b f q  p g b η q  ψ 1 g  p g b θ q  p g b f η q  ψ 1 g  p g b θ f η q  ξ p θ f η q g hene, the follo wing diagram omm utes: Hom C p X, Y q Hom C p η,F θ q / / ξ   Hom C p X 1 , Y 1 q ξ   Hom C Γ p X r Γ s , p Y , ψ qq Hom C Γ p η r Γ s ,θ q / / Hom C Γ p X 1 r Γ s , p Y 1 , ψ 1 qq whi h leads to the funtorialit y of ξ . If f 1 P Hom C Γ p X r Γ s , p Y , ψ qq , let ζ p f 1 q  f 1 1 λ  1 . One learly has ζ p ξ p f qq  f . Moreo v er, ξ p ζ p f 1 qq g  ψ g  p g b p f 1 1 λ  1 qq  ψ g  p g b f 1 1 q  p g b λ  1 q  f 1 g  χ g | 1 b X  p g b λ  1 q  f 1 g  p g b λ q  p g b λ  1 q  f 1 g th us ξ and ζ are reipro al morphisms.  Corollary 2.23. L et p X, ψ q P C Γ . Then (i) The obje t p X, ψ q is a dir e t summand of X r Γ s . (ii) If p X, ψ q is inde  omp osable, Γ ats tr ansitively on the set of iso lasses of inde  omp osable summands of X . Pr o of. (i) By prop osition 2.22 , X r Γ s  k r Γ s b p X, ψ q . This giv es the result b eause 1 is a diret summand of k r Γ s . (ii) Let X  À ℓ i  1 X i in C , the X i b eing indeomp osable. Then X r Γ s  À ℓ i  1 X i r Γ s and, as C Γ is Krull-S hmidt, p X, ψ q is a diret summand of one of the X i r Γ s . As a onsequene, add p X q  add F p X i r Γ sq  add  À g P Γ p g b X i q  . As X i is indeomp osable and C is Krull-S hmidt, Γ ats learly transitiv ely on this sub ategory .  Notation 2.24. Let M P C Γ . The n um b er of indeomp osable diret summands of F M will b e denoted b y ℓ p M q . The n um b er of non isomorphi indeomp osable diret summands of F M will b e denoted b y # M . R emark 2.25 . If M is indeomp osable and Γ is yli, then # M  ℓ p M q . Ho w ev er, it is not the ase in general, ev en if the group is omm utativ e (see prop osition 2.7 and the remark after it). Lemma 2.26. L et X P C Γ b e inde  omp osable. Then, # X divides ℓ p X q . Their r atio is the numb er of  opies of e ah inde  omp osable of add p F X q in F X . Pr o of. Let X 1 , X 2 b e t w o indeomp osable summands of F X . By orollary 2.23 , as X is inde- omp osable, there exists g P Γ su h that g b X 1  X 2 . Moreo v er, g b X  X , therefore, the n um b ers of opies of X 1 and X 2 in X are equal.  Lemma 2.27. If C is semisimple then C Γ is semisimple. 15 Pr o of. Let X, Y t w o indeomp osable ob jets of C Γ and f P Hom C Γ p X, Y q . As C is semisimple, C and C Γ are ab elian. Let k : K ã Ñ X the k ernel of f . As C is semisimple, F k splits through an r h : F X ։ F K . One gets r h P Hom C Γ p X, K q . Let h  1 #Γ ¸ g P Γ g  r h with the result that h is a morphism from X to K and that hk  1 #Γ ¸ g P Γ p g  r h q k  1 #Γ ¸ g P Γ p g  r h qp g  k q  1 #Γ ¸ g P Γ g  p r hk q  1 #Γ ¸ g P Γ g  Id K  Id K and nally , k splits. As C Γ is Krull-S hmidt, K is a diret summand of X . Finally , as X is indeomp osable, k er f  0 or k er f  X . In the same w a y , cok er f  0 or cok er f  Y and, as a onsequene, f  0 or f is in v ertible whi h sho ws that C Γ is semisimple.  Denition 2.28. One denotes b y r C s the semisimple k -ategory , whose simple ob jets are the isomorphism lasses of indeomp osable ob jets of C . F or X  X 1 ` X 2 `    ` X n P C where X 1 , X 2 , . . . , X n are indeomp osable, write r X s  r X 1 s ` r X 2 s `    ` r X n s where r X i s is the isomorphism lass of X i (it is w ell dened b eause C is Krull-S hmidt). More- o v er, if X P C is indeomp osable, one denes End r C s pr X sq  End C p X q{ m where m is the maximal ideal of End C p X q . Lemma 2.29. L et X 1 , X 2 , . . . , X n P C b e non isomorphi inde  omp osable obje ts. Then, for i 1 , i 2 , . . . , i n , j 1 , j 2 , . . . , j n P N , dim k p Hom r C s pr X i 1 1 ` X i 2 2 `    ` X i n n s , r X j 1 1 ` X j 2 2 `    ` X j n n sqq  n ¸ ℓ  1 i ℓ j ℓ c ℓ . Her e, for every ℓ , c ℓ is the de gr e e of the extension k  End C p X ℓ q{ m ℓ and m ℓ denotes the maximal ide al of End C p X ℓ q . Pr o of. It is ob vious.  Lemma 2.30. (i) The ation of Γ on C indu es an ation of Γ on r C s ; (ii) if k is algebr ai al ly lose d and r C s has only one simple, then r C s Γ  r C Γ s as a mo d k r Γ s - mo dule  ate gory. Pr o of. First of all, up to an equiv alene of ategories, ea h isomorphism lass of C an b e supp osed to on tain exatly one ob jet. (i) If g P Γ and X P C , let g b r X s  r g b X s . If f : r X s Ñ r X s is a morphism, then f omes from a morphism f 0 : X Ñ X . Let g b f b e the lass of g b f 0 mo dulo m . It is w ell dened as if f 0 is nilp oten t, then g b f 0 is also nilp oten t. In the same w a y , one denes strutural morphisms of r C s b y pro jeting those of C mo dulo maximal ideals. It is no w lear that r C s is a Γ -mo dule ategory . (ii) Let X 0 P C b e the only indeomp osable ob jet up to isomorphism. As r C s Γ and r C Γ s are semisimple and k is algebraially losed, it is enough to see that the simple ob- jets of b oth ategories are in bijetion. If rp X, ψ qs P r C Γ s , one an asso iate to it pr X s , ψ 1 q P r C s Γ b y reduing matrix o eien ts of ψ mo dulo the maximal ideal of End C p X 0 q . Con v ersely , if pr X s , ψ 1 q P r C s Γ , one ar asso iate to it rp X, ψ qs where the matrix o eien ts of ψ are those of ψ 1 m ultiplied b y Id X 0 . It is learly a bijetion.  16 Lemma 2.31. Supp ose that k is algebr ai al ly lose d. L et X P C b e an inde  omp osable obje t and Y P add p X r Γ sq an inde  omp osable obje t of C Γ . Then X r Γ s has ℓ p Y q{ # p Y q inde  omp osable dir e t summands isomorphi to Y . Pr o of. It will b e pro v ed in three steps. (i) # Y  1 . Then, up to restrition to add p F Y q  add p X q , one an use lemma 2.30 and dim Hom r C Γ s pr Y s , r k r Γ s b Y sq  dim Hom r C s Γ pr Y s , k r Γ s b r Y sq  dim Hom r C s p F r Y s , F r Y sq  ℓ p Y q 2 . Moreo v er, X r Γ s ℓ p Y q  k r Γ s b Y whi h implies the result in this ase. (ii) # Y  #Γ . In this ase, X r Γ s is indeomp osable and therefore Y  X r Γ s whi h learly implies the result. (iii) General ase. Let r Y b e the set of indeomp osable summands of F Y up to isomorphism. The ation of Γ indues a morphism Γ Ñ S r Y , whose k ernel H is normal in Γ . So, one an apply prop osition 2.11 . As C Γ  C H p Γ { H q , one gets a partial forgetful funtor from C Γ to C H . Let r Y P C H b e the image of Y b y this funtor. Let Y 1 b e an indeomp osable diret summand of r Y . As # Y 1  1 , ase (i ) applies to Y 1 . Moreo v er, X r Γ s  X r H sr Γ { H s and # Y  #Γ { H . Hene, Y  Y 1 r Γ { H s app ears in X r Γ s the n um b er of times Y 1 app ears in X r H s , that is ℓ p Y 1 q  ℓ p Y q{ # p Γ { H q  ℓ p Y q{ # Y .  2.5. Appro ximations. Denition 2.32. One denes Add p C q to b e the lass of all full sub- k -ategories of C whi h are stable under isomorphisms and diret summands. If E is a olletion of ob jets of C , one denotes b y add p E q the smallest ategory of Add p C q on taining E . The ategory T P Add p C q is said to b e nitely generated if T is of the form add p M q for some ob jet M P C . The sub lass of Add p C q onsisting in all nitely generated ategories will b e denoted b y add p C q . If F is a funtor from C to a k -ategory C 1 and T P Add p C q , F p T q will denote add pt F p X q | X P T uq . If C is an M -mo dule ategory for some monoidal ategory M , Add p C q M (resp. add p C q M ) will denote the lass of elemen ts of Add p C q (resp. add p C q ) whi h are sub- M -mo dule ategories of C . Denition 2.33. Let T P Add p C q and M P C . A left (r esp. right) T -appr oximation of M is an ob jet N P T and a morphism f P Hom C p M , N q (resp. P Hom C p N , M q ) su h that for ev ery N 1 P T and f 1 : M Ñ N 1 (resp. f 1 : N 1 Ñ M ), f 1 fators through f . Denition 2.34. A morphism f of the ategory C is said to b e left (r esp. right) minimal if ev ery morphism g su h that f  g  f (resp. f  f  g ) is an isomorphism. The follo wing lemmas are folklore. F or detailed pro ofs, the reader is referred to [ ARS ℄ or [Dem3 ℄. Lemma 2.35. L et X, Y , X 1 , Y 1 P C and f P Hom C p X, Y q , f 1 P Hom C p X 1 , Y 1 q . (i) The morphism f is right minimal if and only if ther e is no de  omp osition X  X 0 ` X 1 suh that f | X 0  0 and X 0  0 . (ii) The morphism f is left minimal if and only if ther e is no de  omp osition Y  Y 0 ` Y 1 suh that the  or estrition of f to Y 0 vanishes and Y 0  0 . (iii) The morphisms f and f 1 ar e b oth left (r esp. right) minimal if and only if f ` f 1 is left (r esp. right) minimal. Lemma 2.36. L et M , N , M 1 , N 1 P C , T P Add p C q , f : M Ñ N and f 1 : M 1 Ñ N 1 . Then f and f 1 ar e b oth left (r esp. right) T -appr oximation if and only if f ` f 1 is a left (r esp. right) T -appr oximation. Lemma 2.37. If T P add p C q and M P C , then ther e exists a minimal left (r esp. right) T - appr oximation of M whih is unique up to (non unique) isomorphism. Mor e over, any left (r esp. right) T -appr oximation of M is the dir e t sum of the minimal left (r esp. right) T -appr oximation of M and a morphism of the form 0 Ñ N (r esp. N Ñ 0 ). 17 Lemma 2.38. L et M P C and T P Add p C q . If T  ontains the inje tive envelop e (r esp. pr oje tive  over) of M , then every left (r esp. right) T -appr oximation of M is an admissible monomorphism (r esp. epimorphism). Lemma 2.39. L et T P Add p C Γ q mod k r Γ s , X, Y P C Γ and f P Hom C Γ p X, Y q . (i) f is a left (r esp. right) T -appr oximation if and only if F f is a left (r esp. right) F T - appr oximation. (ii) f is a minimal left (r esp. right) T -appr oximation if and only if F f is a minimal left (r esp. right) F T -appr oximation. Pr o of. By dualit y , it is enough to pro v e the statemen t for left appro ximations. (i) First of all, as T is mo d k r Γ s -stable, if F Y P F T , then Y P T (and, b y denition, if Y P T , F Y P F T ). As F and r Γ s are adjoin t, the follo wing diagram is omm utativ e for ev ery T P T : Hom C p F Y , F T q Hom C p F f, F T q / /    Hom C p F X , F T q    Hom C Γ p Y , p F T qr Γ sq Hom C Γ p f , p F T qr Γ sq / / Hom C Γ p X, p F T qr Γ sq . The rst line is surjetiv e for ev ery T if and only if F f is a left F T -appro ximation. As T is mo d k r Γ s -stable, p F T qr Γ s  k r Γ s b T P T and therefore the seond line is surjetiv e for ev ery T if and only if f is a left T -appro ximation (b eause T is a diret summand of p F T qr Γ s ). (ii) If F f is left minimal then f is learly left minimal. Con v ersely , supp ose that f is a minimal left T -appro ximation. Let r f : F X Ñ r Y b e a minimal left F T -appro ximation of F X . Let f 1  r f r Γ s : p F X qr Γ s Ñ r Y r Γ s . Then, one gets F f 1  à g P Γ g b r f . F or ev ery g P Γ , g b r f is a minimal left T -appro ximation b eause F T is Γ -stable, and, as a onsequene, b y using lemmas 2.36 and 2.35 , F f 1 is also a minimal left F T - appro ximation. Then, f 1 is a minimal left T -appro ximation. By uniit y of a minimal left T -appro ximation, and using lemmas 2.36 and 2.35 , f 1  f ` g where g is a minimal left T -appro ximation of À g P Γ zt e u g b X . Finally , as F f 1  F f ` F g is left minimal, F f is also left minimal b y lemma 2.35 .  2.6. A tion on an exat ategory. The ation of Γ on C is no w supp osed to b e exat. Hene, C Γ is exat. It is easy to see that the funtor r Γ s from C to C Γ is exat. Lemma 2.40. If X P C is inje tive (r esp. pr oje tive), then X r Γ s is inje tive (r esp. pr oje tive). Pr o of. Supp ose that X is injetiv e. Let 0 Ñ X r Γ s f Ý Ñ Y g Ý Ñ Z Ñ 0 b e an admissible short exat sequene in C Γ . By denition, 0 Ñ F p X r Γ sq f Ý Ñ F Y g Ý Ñ F Z Ñ 0 is an admissible short exat sequene in C . Applying Hom C Γ p , X r Γ sq and Hom C p , X q giv es the long exat sequenes 0 Ñ Hom C Γ p Z, X r Γ sq Ñ Hom C Γ p Y , X r Γ sq Ñ Hom C Γ p X r Γ s , X r Γ sq Ñ . . . 0 Ñ Hom C p F Z , X q Ñ Hom C p F Y , X q Ñ Hom C p F p X r Γ sq , X q Ñ Ext 1 C p F Z , X q  0 . The isomorphism of bifuntors Hom C Γ p , r Γ sq  Hom C p F  , q p ermits to onlude that Hom C Γ p f , X r Γ sq : Hom C Γ p Y , X r Γ sq Ñ Hom C Γ p X r Γ s , X r Γ sq is surjetiv e. Therefore the admissi- ble short exat sequene 0 Ñ X r Γ s f Ý Ñ Y g Ý Ñ Z Ñ 0 splits. Hene, X r Γ s is injetiv e. The pro of is similar for the pro jetiv e ase.  18 Corollary 2.41. If C has enough inje tive (r esp. pr oje tive) obje ts, then C Γ has also enough inje tive (r esp. pr oje tive) obje ts. Mor e over, for X P C Γ , ther e exists an inje tive r esolution I  (r esp. a pr oje tive r esolution P  ) of X suh that F I  (r esp. F P  ) is an inje tive r esolution (r esp. a pr oje tive r esolution) of F X . Pr o of. Supp ose that C has enough injetiv e ob jets. Let X P C Γ . There exists an admissible short exat sequene 0 Ñ F X Ñ I Ñ Y Ñ 0 in C where I is injetiv e. As the ation of Γ is exat, it giv es an admissible short exat sequene 0 Ñ p F X qr Γ s Ñ I r Γ s Ñ Y r Γ s Ñ 0 . Moreo v er, X is a diret summand of p F X qr Γ s and, as the omp osition of t w o admissible monomorphisms is an admissible monomorphism, one gets an admissible monomorphism X Ñ I r Γ s . F or the seond part, it is enough to apply indutiv ely the rst part b eause I r Γ s and F p I r Γ sq are b oth injetiv e in C Γ and C , resp etiv ely .  One supp oses no w that C has enough injetiv es or enough pro jetiv es. Denition 2.42. F or X P C Γ and n P N , Ext n C Γ p X, q will denote the righ t deriv ed funtor of Hom C Γ p X, q if C has enough injetiv e ob jets and Ext n C Γ p , X q will denote the left deriv ed funtor of Hom C Γ p , X q if C has enough pro jetiv e ob jets. If C has b oth enough injetiv e and pro jetiv e ob jets, the t w o denitions oinide as usual. Hene, one gets on C Γ a struture of mo d k r Γ s -exat ategory . All usual homologial results remain true in this on text. The follo wing prop osition links these prop erties with the usual k -exat struture and summarizes some prop erties of Ext . Prop osition 2.43. F or every n P N , ther e ar e funtorial isomorphisms (the ? ar e variables in mo d k r Γ s and the  ar e variables in C Γ ): (i) Ext n C Γ p ? 1 b  1 , ? 2 b  2 q  ?  1 b ? 2 b Ext n C Γ p 1 ,  2 q ; (ii) Ext n C Γ p ? 1 b  1 , ? 2 b  2 q  Hom mod k r Γ s p ? 1 b ?  2 , Ext n C Γ p 1 ,  2 qq ; (iii) F Ext n C Γ p , q  Ext n C p F  , F q ; (iv) Ext n C Γ p ?  b  , q  Ext n C Γ p , ? b q ; (v) Ext n C Γ p ? r Γ s , q  Ext n C p ? , F q ; (vi) Ext n C Γ p , ? r Γ sq  Ext n C p F  , ? q wher e F denotes the for getful funtors fr om C Γ to C and fr om mo d k r Γ s to mo d k . Mor e over, these isomorphisms  ommute with long exat se quen es obtaine d fr om admissible short exat se quen es in C Γ . Pr o of. These are easy onsequenes of lemma 2.41 together with prop osition 2.17 and standard homologial onstrutions.  Corollary 2.44. If X P C Γ , X is inje tive (r esp. pr oje tive) if and only if F X is inje tive (r esp. pr oje tive). Pr o of. If X is injetiv e, for ev ery Y P C , Ext n C p Y , F X q  Ext n C Γ p Y r Γ s , X q  0 so that F X is injetiv e. If F X is injetiv e, for all Y P C Γ , as Y is a diret summand of p F Y qr Γ s , Ext n C Γ p Y , X q  Ext n C Γ pp F Y qr Γ s , X q  Ext n C p F Y , F X q  0 hene X is injetiv e. The pro of is the same for the pro jetiv e ase.  Lemma 2.45. F or every r epr esentation r P mo d k r Γ s , the funtor r b  fr om C Γ to C Γ is exat. Pr o of. Let 0 Ñ X Ñ Y Ñ Z Ñ 0 b e an admissible short exat sequene of C Γ . By denition, 0 Ñ F X Ñ F Y Ñ F Z Ñ 0 is an admissible short exat sequene of C . As 0 Ñ F p r b X q Ñ F p r b Y q Ñ F p r b Z q Ñ 0 is isomorphi to 0 Ñ p F X q dim r Ñ p F Y q dim r Ñ p F Z q dim r Ñ 0 , this is an admissible short exat sequene of C and therefore, b y denition, 0 Ñ r b X Ñ r b Y Ñ r b Z Ñ 0 is an admissible short exat sequene of C Γ .  19 2.7. A tion on a F rob enius stably 2 -Calabi-Y au ategory. As b efore, the group Γ is supp osed to b e nite, of ardinalit y non divisible b y the  harateristi of k and the k -ategory C is exat, Hom -nite, Krull-S hmidt. Reall that C is alled F r ob enius if it has enough pro jetiv es and enough injetiv es and if the pro jetiv e ob jets and the injetiv e ob jets are the same. Reall that C is alled (stably) 2 -Calabi-Y au if there is a funtorial isomorphism c : Ext 1 C p 1 ,  2 q  Ext 1 C p 2 ,  1 q  . In the follo wing, C will b e supp osed to b e F rob enius and stably 2 -Calabi- Y au. The ategory C Γ is F rob enius b y orollary 2.44 . One xes an isomorphism of bifuntors c : Ext 1 C p 1 ,  2 q  Ext 1 C p 2 ,  1 q  . Denition 2.46. The ation of Γ on C is said to b e 2 -Calabi-Y au (for c ) if it is exat and for ev ery g P Γ , the follo wing diagram omm utes: Ext 1 C p 1 ,  2 q c   g b / / Ext 1 C p g b  1 , g b  2 q c   Ext 1 C p 2 ,  1 q  Ext 1 C p g b  2 , g b  1 q  p g bq  o o F rom no w on, the ation of Γ on C is assumed to b e 2 -Calabi-Y au (for c ). Prop osition 2.47. (i) The funtorial isomorphism of ve tor sp a es c is also a funtorial isomorphism of k r Γ s -mo dules: Ext 1 C Γ p 1 ,  2 q  Ext 1 C Γ p 2 ,  1 q  (r e  al l that for any X, Y P C Γ , the underlying ve tor sp a e of the mo d k r Γ s -mo dule Ext 1 C Γ p X, Y q is Ext 1 C p F X , F Y q ). (ii) The  ate gory C Γ is 2 -Calabi-Y au. Pr o of. Reall that there is an isomorphism of funtors from mo d k r Γ s in to itself: Hom mod k r Γ s p 1 , q  Hom mod k r Γ s p , 1 q  . (i) The only thing to pro v e is that, for an y p X, ψ q , p Y , χ q P C Γ , c X,Y is in fat a morphism of represen tations of Γ from Ext 1 C Γ pp X, ψ q , p Y , χ qq to Ext 1 C Γ pp Y , χ q , p X, ψ qq  . F or g P Γ , it is enough to sho w that the follo wing diagram omm utes: Ext 1 C p X, Y q c X,Y / / g b   Ext 1 C p Y , X q  pp g bq  q  1   Ext 1 C p g b X, g b Y q c g b X, g b Y / / Ext 1 C p ψ  1 g ,χ g q   Ext 1 C p g b Y , g b X q   Ext 1 C p χ g ,ψ  1 g q  p Ext 1 C p χ  1 g ,ψ g q  q  1   Ext 1 C p X, Y q c X,Y / / Ext 1 C p Y , X q  (the left side omes from the ation of g on Ext 1 C p X, Y q and the righ t side omes from the in v erse of the adjoin t of the ation of g ); the upp er square omm utes b eause the ation of Γ is 2 -Calabi-Y au and the lo w er square omm utes b eause the isomorphism c is funtorial. (ii) Denote b y c the isomorphism of funtors of (i ). Let r c  Hom mod k r Γ s p 1 , c q . Then r c is an isomorphism of funtors Hom mod k r Γ s p 1 , Ext 1 C Γ p 1 ,  2 qq Ý Ñ Hom mod k r Γ s p 1 , Ext 1 C Γ p 2 ,  1 q  q . The reipro al isomorphism is r c  1  Hom mod k r Γ s p 1 , c  1 q . 20 Moreo v er, lemma 2.43 leads to Hom mod k r Γ s p 1 , Ext 1 C Γ p 1 ,  2 qq  Ext 1 C Γ p 1 ,  2 q and Hom mod k r Γ s p 1 , Ext 1 C Γ p 2 ,  1 q  q  Hom mod k r Γ s p Ext 1 C Γ p 2 ,  1 q , 1 q  Hom mod k r Γ s p 1 , Ext 1 C Γ p 2 ,  1 qq   Ext 1 C Γ p 2 ,  1 q  whi h nishes the pro of.  2.8. Computation of p k Q q Γ and p Λ Q q Γ . The aim of this setion is to summarize some prop- erties pro v ed in [Dem1℄ useful to ompute equiv arian t ategories for ategories of mo dules. The assumptions on k and Γ are the same as b efore. If Λ is a k -algebra and if Γ ats on Λ , the ation b eing denoted exp onen tially , the sk ew group algebra of Λ under the ation of Γ is b y denition the k -algebra whose underlying k -v etor spae is k r Γ s b k Λ and whose m ultipliation is linearly generated b y p g b a qp g 1 b a 1 q  g g 1 b a g 1 1 a 1 for all g , g 1 P Γ and a, a 1 P Λ (see [RR ℄). It will b e denoted b y ΛΓ . Iden tifying k r Γ s and Λ with subalgebras of ΛΓ , an alternativ e denition is ΛΓ  x Λ , k r Γ s | p g , a q P Γ  Λ , g ag  1  a g y k -alg The follo wing links sk ew group algebras with equiv arian t ategories. Prop osition 2.48. The ation of Γ on Λ indu es an ation of Γ on mo d Λ . Mor e over, ther e is a  anoni al e quivalen e of  ate gories b etwe en mo d p ΛΓ q and mo d p Λ q Γ . Pr o of. If g P Γ and p V , r q P mo d Λ , one denotes b y g b p V , r q the represen tation p V , g b r q of Λ where, if a P Λ , p g b r q a  r g  1 a . If f P Hom mod Λ pp V , r q , p V 1 , r 1 qq , one denes Id g b f  f . Extending this denition b y linearit y on the whole ategory Γ , mo d Λ is a Γ -mo dule ategory . The strutural isomorphisms are iden tities. Let p V , r , ψ q P mo d p Λ q Γ . F or g P Γ and a P Λ , one denes r ψ g b a  ψ g r a . Sine for ev ery g , g 1 P Γ and a, a 1 P Λ the diagram V r a 1 / / r g 1 1 p a q a 1 ) ) S S S S S S S S S S S S S S S S S S S V ψ g 1 / / r g 1 1 p a q p g 1 b r q a   V r a   V ψ gg 1 ) ) S S S S S S S S S S S S S S S S S S S ψ g 1 / / V ψ g   V is omm utativ e, one gets that r ψ g b a r ψ g 1 b a 1  r ψ p g b a qp g 1 b a 1 q . Therefore p V , r ψ q is a represen tation of ΛΓ . If f P Hom mod p Λ q Γ pp V , r , ψ q , p V 1 , r 1 , ψ 1 qq , it is lear that f P Hom mod p ΛΓ q pp V , r ψ q , p V 1 , r 1 ψ 1 qq . Hene p V , r , ψ q ÞÑ p V , r ψ q is a funtor from mo d p Λ q Γ to mo d p ΛΓ q . Let p V , r 0 q P mo d p ΛΓ q . If a P Λ , let r a  r 0 1 b a whi h giv es that p V , r q P mo d p Λ q . If g P Γ , let ψ g  r 0 g b 1 . F or g P Γ and a P Λ , the follo wing diagram omm utes: V ψ g / / r g  1 p a q   r 0 g b g  1 p a q & & N N N N N N N N N N N N N V r a   V ψ g / / V . 21 Hene ψ g is an isomorphism from g b p V , r q to p V , r q . It is lear that p V , r , ψ q P mo d p Λ q Γ . If f P Hom mod p ΛΓ q pp V , r 0 q , p V 1 , r 1 0 qq , one gets immediately f P Hom mod p Λ q Γ pp V , r , ψ q , p V 1 , r 1 , ψ 1 qq . The t w o onstruted funtors are m utually in v erse.  Let no w Q  p Q 0 , Q 1 q b e a quiv er. Consider an ation of Γ on the path algebra k Q p erm uting the set of primitiv e idemp oten ts t e i | i P Q 0 u . W e no w dene a new quiv er Q Γ . Let r Q 0 b e a set of represen tativ es of the lasses of Q 0 under the ation of Γ . F or i P Q 0 , let Γ i denote the subgroup of Γ stabilizing e i , let i  P r Q 0 b e the represen tativ e of the lass of i and let κ i P Γ b e su h that κ i i   i . F or p i, j q P r Q 2 0 , Γ ats on O i  O j where O i and O j are the orbits of i and j under the ation of Γ . A set of represen tativ es of the lasses of this ation will b e denoted b y F ij . F or i, j P Q 0 , dene A ij  e j  rad p k Q q{ rad p k Q q 2  e i where rad p k Q q is the Jaobson radial of k Q . W e regard A ij as a left k r Γ i X Γ j s -mo dule b y restriting the ation of Γ . The quiv er Q Γ has v ertex set Q Γ , 0  ¤ i P r Q 0 t i u  irr p Γ i q where irr p Γ i q is a set of represen tativ es of isomorphism lasses of irreduible represen tations of Γ i . The set of arro ws of Q Γ from p i, ρ q to p j, σ q is a basis of à p i 1 ,j 1 qP F ij Hom mod k r Γ i 1 X Γ j 1 s pp κ i 1  ρ q | Γ i 1 X Γ j 1 b A i 1 j 1 , p κ j 1  σ q | Γ i 1 X Γ j 1 q  where the represen tation κ i 1  ρ of Γ i 1 is the same as ρ as a v etor spae, and p κ i 1  ρ q g  ρ κ  1 i 1 g κ i 1 for g P Γ i 1  κ i 1 Γ i κ  1 i 1 . Theorem 2.49 ([Dem1 , theorem 1℄) . Ther e is an e quivalen e of  ate gories mo d k p Q Γ q  mo d p k Q q Γ . Theorem 2.49 w as also pro v ed b y Reiten and Riedtmann in [RR, 2℄ for yli groups. The follo wing theorem deals with the ase of prepro jetiv e algebras Λ Q . Theorem 2.50 ([Dem1 , theorem 2℄) . If Γ ats on k Q , wher e Q is the double quiver of Q , by p ermuting the primitive idemp otents e i , and if for al l g P Γ , r g  r wher e r is the pr epr oje tive r elation of this quiver, then  Q  Γ is of the form Q 1 for some quiver Q 1 and p Λ Q q Γ is Morita e quivalent to Λ Q 1 . One an alw a ys extend an ation on k Q to an ation on k Q and this yields: Corollary 2.51 ([Dem1 , orollary 1℄) . A n ation of Γ on a p ath algebr a k Q p ermuting the primitive idemp otents indu es natur al ly an ation of Γ on k Q and  Q  Γ is isomorphi to the double quiver of Q Γ . Mor e over, ther e is an e quivalen e of  ate gories mo d Λ Q Γ  mo d Λ Q Γ . 3. Ca tegorifia tion of skew-symmetrizable luster algebras In this part, C is supp osed to b e exat, F rob enius, Hom -nite, stably 2 -Calabi-Y au and Krull- S hmidt. As b efore, Γ is a nite group whose ardinalit y is not divisible b y the  harateristi of k . The group Γ is supp osed to at on C , the ation b eing exat and 2 -Calabi-Y au (see denition 2.46 ). The results of this setion generalize w orks b y Geiÿ, Leler and S hrö er (in partiular [ GLS3 ℄ and [GLS1 ℄) in the on text of prepro jetiv e algebras, and w orks of Deh y , F u, Keller, P alu and others in the on text of exat ategories (see [DK℄, [FK ℄, [P al ℄). 22 3.1. Mutation of maximal Γ -stable rigid sub ategories. Denition 3.1. A ategory T P Add p C q (resp. P Add p C Γ q ) is said to b e rigid if there are no non trivial extensions b et w een its ob jets. If moreo v er ev ery rigid ategory T 1 on taining T is equal to T , then w e sa y that T is maximal rigid . The follo wing denition w as in tro dued in [ Iy a2℄: Denition 3.2. A ategory T P Add p C q (resp. P Add p C Γ q ) is said to b e luster-tilting or maximal 1 -ortho gonal if for an y X P C (resp P C Γ ), the follo wing are equiv alen t:   Y P T , Ex t 1 p X, Y q  0 ;  X P T . Clearly , an y luster-tilting ategory is maximal rigid. Denition 3.3. If T P Add p C q Γ is rigid, T is said to b e maximal Γ -stable rigid if for ev ery rigid T 1 P Add p C q Γ on taining T , T 1  T . Similarly , if T P Add p C Γ q mod k r Γ s is rigid, T is said to b e maximal mo d k r Γ s -stable rigid if for ev ery rigid T 1 P Add p C Γ q mod k r Γ s on taining T , T 1  T . R emark 3.4 . Being maximal Γ -stable (resp. mo d k r Γ s -stable) rigid is w eak er than b eing maximal rigid and Γ -stable (resp. mo d k r Γ s -stable). F or instane, the quiv er 1 α ( ( 2 α  h h with relations αα   α  α  0 , on whi h Z { 2 Z ats b y ex hanging the t w o v erties, has no maximal rigid ob jet whi h is Z { 2 Z -stable. The only maximal Γ -stable rigid ob jet is the diret sum of the pro jetiv e ones. Prop osition 3.5. The funtors r Γ s and F indu e r e ipr o  al bije tions b etwe en: (i) Add p C q Γ and Add p C Γ q mod k r Γ s ; (ii) the set of rigid T P Add p C q Γ and the set of rigid T P Add p C Γ q mod k r Γ s ; (iii) the set of maximal Γ -stable rigid T P Add p C q Γ and the set of maximal mo d k r Γ s -stable rigid T P Add p C Γ q mod k r Γ s ; (iv) the set of luster-tilting T P Add p C q Γ and the set of luster-tilting T P Add p C Γ q mod k r Γ s . Mor e over, al l these bije tions r estrit to bije tions b etwe en the  orr esp onding nitely gener ate d lasses. Pr o of. (i) If D P Add p C q Γ , it is lear that D r Γ s P Add p C Γ q mod k r Γ s . Similarly , if D 1 P Add p C Γ q mod k r Γ s , it is easy to see that F D 1 P Add p C q Γ . Supp ose no w that X P F p D r Γ sq . This means that X is a diret summand of F p Y r Γ sq for some Y P D . But F p Y r Γ sq is the diret sum of the g b Y for g P Γ , hene F p Y r Γ sq P D and nally , X P D . If X P D , X is a diret summand of F p X r Γ sq so that X is in F p D r Γ sq . One onludes that F p D r Γ sq  D . Supp ose that X P F p D 1 qr Γ s . It means that X is a diret summand of some F p Y qr Γ s  k r Γ s b Y where Y P D 1 and, as D 1 is mo d k r Γ s -stable, F p Y qr Γ s and X are in D 1 . On the other hand, if X P D 1 , as X is a diret summand of F p X qr Γ s , X is in F p D 1 qr Γ s . Finally F p D 1 qr Γ s  D 1 . Therefore, F and r Γ s indue reipro al bijetions. (ii) Supp ose that T P Add p C q Γ is rigid. Let X P T r Γ s . By denition, there exists Y P T and X 1 P T r Γ s su h that Y r Γ s  X ` X 1 . Th us, one gets Ext 1 C Γ p X, X q  Ext 1 C Γ p Y r Γ s , Y r Γ sq  Ext 1 C p Y , F p Y r Γ sqq  Ext 1 C p F p Y r Γ sq , F p Y r Γ sqq  0 b eause, as Y P T and T is Γ -stable, F p Y r Γ sq P T . As a onsequene, X is rigid and therefore T r Γ s is rigid. 23 Supp ose no w that T P Add p C Γ q mod k r Γ s is rigid. Let X P F T . By denition, there exists Y P T and X 1 P F T su h that F Y  X ` X 1 . One gets Ext 1 C p X, X q  Ext 1 C p F Y , F Y q  Ext 1 C Γ p Y , p F Y qr Γ sq  Ext 1 C Γ pp F Y qr Γ s , p F Y qr Γ sq  0 b eause, as Y P T and T is mo d k r Γ s -stable, p F Y qr Γ s  k r Γ s b Y P T . As a onsequene, F T is rigid. (iii) This is lear b eause the t w o bijetions are inreasing (with resp et to inlusion). (iv) Supp ose that T P Add p C q Γ is luster-tilting. Let X P C Γ su h that for ev ery Y P T r Γ s , Ext 1 C Γ p X, Y q  0 . In partiular, for ev ery Z P T , Ext 1 C p F X , Z q  Ext 1 C Γ p X, Z r Γ sq  0 . Therefore, as T is luster-tilting, F X P T , p F X qr Γ s P T r Γ s , and X P T r Γ s b eause X is a diret summand of p F X qr Γ s . Finally , as T r Γ s is rigid, T r Γ s is luster-tilting. Supp ose that T P Add p C Γ q mod k r Γ s is luster-tilting. Let X P C b e su h that for ev ery Y P F T , Ext 1 C p X, Y q  0 . In partiular, for all Z P T , Ext 1 C Γ p X r Γ s , Z q  Ext 1 C p X, F Z q  0 . Therefore X r Γ s P T , F p X r Γ sq P F T and, X P F T as X is a diret summand of F p X r Γ sq . Finally , as F T is rigid, F T is luster-tilting.  Lemma 3.6. L et T P Add p C Γ q mod k r Γ s b e rigid, and let X P C Γ b e suh that k r Γ s b X is rigid. If 0 Ñ X f Ý Ñ T g Ý Ñ Y Ñ 0 is an admissible short exat se quen e and f is a left T -appr oximation, then the  ate gory add p T , k r Γ s b Y q is rigid. Pr o of. If T 1 P T , applying Hom C Γ p , k r Γ s b T 1 q to the admissible short exat sequene yields the long exat sequene 0 Ñ Hom C Γ p Y , k r Γ s b T 1 q Ñ Hom C Γ p T , k r Γ s b T 1 q Hom C Γ p f ,k r Γ sb T 1 q Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ñ Hom C Γ p X, k r Γ s b T 1 q Ñ Ext 1 C Γ p Y , k r Γ s b T 1 q Ñ Ext 1 C Γ p T , k r Γ s b T 1 q  0 . As f is a left T -appro ximation, Hom C Γ p f , k r Γ s b T 1 q is surjetiv e, and as a onsequene, Ext 1 C Γ p Y , k r Γ s b T 1 q  Ext 1 C Γ p k r Γ s b Y , T 1 q  0 . Moreo v er, as k r Γ s b  is an exat funtor, 0 Ñ k r Γ s b X Ñ k r Γ s b T k r Γ sb g Ý Ý Ý Ý Ñ k r Γ s b Y Ñ 0 is an admissible short exat sequene. There- fore, applying the funtor Hom C Γ p X, q giv es rise to the long exat sequene 0 Ñ Hom C Γ p X, k r Γ s b X q Ñ Hom C Γ p X, k r Γ s b T q Hom C Γ p X,k r Γ sb g q Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ñ Hom C Γ p X, k r Γ s b Y q Ñ Ext 1 C Γ p X, k r Γ s b X q  0 b eause k r Γ s b X is rigid. F urthermore, applying Hom C Γ p , k r Γ s b Y q to the rst admissible short exat sequene yields the long exat sequene 0 Ñ Hom C Γ p Y , k r Γ s b Y q Ñ Hom C Γ p T , k r Γ s b Y q Hom C Γ p f ,k r Γ sb Y q Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ñ Hom C Γ p X, k r Γ s b Y q Ñ Ext 1 C Γ p Y , k r Γ s b Y q Ñ Ext 1 C Γ p T , k r Γ s b Y q  0 Let no w h P Hom C Γ p X, k r Γ s b Y q . By the previous argumen t, h fatorizes through k r Γ s b g . Let h 1 P Hom C Γ p X, k r Γ s b T q b e su h that h  p k r Γ s b g q h 1 . As k r Γ s b T P T and f is a left T -appro ximation, there exists t P Hom C Γ p T , k r Γ s b T q su h that h 1  tf . Hene, h  p k r Γ s b g q tf and Hom C Γ p f , k r Γ s b Y q is surjetiv e. Therefore, Ext 1 C Γ p Y , k r Γ s b Y q  0 and T ` k r Γ s b Y is rigid. Finally , add p T , k r Γ s b Y q is rigid.  Prop osition 3.7. L et T P add p C Γ q mod k r Γ s rigid, and let X P C Γ b e inde  omp osable suh that X R T . Supp ose that T  ontains al l pr oje tive obje ts of C Γ , and that add p T , k r Γ s b X q is rigid. Then, ther e exist two admissible short exat se quen es whih ar e unique up to isomorphism 0 Ñ X f Ý Ñ T g Ý Ñ Y Ñ 0 and 0 Ñ Y 1 f 1 Ý Ñ T 1 g 1 Ý Ñ X Ñ 0 24 suh that (i) f and f 1 ar e minimal left T -appr oximations; (ii) g and g 1 ar e minimal right T -appr oximations; (iii) add p T , k r Γ s b Y q and add p T , k r Γ s b Y 1 q ar e rigid; (iv) Y R T and Y 1 R T ; (v) Y and Y 1 ar e inde  omp osable; (vi) add p k r Γ s b X q X add p k r Γ s b Y q  0 and add p k r Γ s b X q X add p k r Γ s b Y 1 q  0 . Pr o of. By symmetry , it is enough to pro v e the results for the rst admissible short exat sequene. Using lemma 2.38 , a minimal left T -appro ximation of X is an admissible monomorphism, whi h implies the existene and the uniit y of the admissible short exat sequene. Then (i) is satised b y denition. As X R T , the admissible short exat sequene do es not split. Hene, Y R add p T , k r Γ s b X q whi h is rigid. This pro v es (iv) and (vi). Moreo v er, lemma 3.6 sho ws that add p T , k r Γ s b Y q is rigid. Hene, (iii ) is pro v ed. F or T 1 P T , applying Hom C Γ p T 1 , q to the admissible short exat sequene yields the long exat sequene 0 Ñ Hom C Γ p T 1 , X q Ñ Hom C Γ p T 1 , Y q Hom C Γ p T 1 ,g q Ý Ý Ý Ý Ý Ý Ý Ý Ñ Hom C Γ p T 1 , Z q Ñ Ext 1 C Γ p T 1 , X q  0 and, therefore, Hom C Γ p T 1 , g q is surjetiv e, whi h implies that g is a righ t T -appro ximation. Let T 0 b e a diret summand of T on whi h g v anishes. Let π b e a pro jetion on T 0 . As g π  0 and f is a k ernel of g , there exists f 1 P Hom C Γ p T , X q su h that π  f f 1 . Therefore, f f 1 p Id T  π q  0 and, as f is an admissible monomorphism, f 1 p Id T  π q  0 . Th us p f 1 f q 2  f 1 π f  f 1 f . As ev ery idemp oten t splits and X is indeomp osable, f 1 f  0 or f 1 f  Id X . As the admissible short exat sequene do es not split, f 1 f  0 . Finally , π  f f 1  p f f 1 q 2  0 so that T 0  0 and g is minimal. This pro v es ( ii). Supp ose no w that Y  Y 1 ` Y 2 . Let g 1 : T 1 Ñ Y 1 and g 2 : T 2 Ñ Y 2 b e minimal righ t T -appro ximations. Then g 1 ` g 2 is a minimal righ t T -appro ximation using lemmas 2.35 and 2.36 . By uniit y of minimal appro ximations, g  g 1 ` g 2 . Therefore, as g is an admissible epimorphism, g 1 and g 2 are admissible epimorphisms. Hene they ha v e k ernels f 1 : X 1 Ñ T 1 and f 2 : X 2 Ñ T 2 and b y uniit y of the k ernel, f  f 1 ` f 2 . As a onsequene, as X  X 1 ` X 2 is indeomp osable, X 1  0 or X 2  0 and as f is minimal, T 1  0 or T 2  0 . So, Y 1  0 or Y 2  0 . Finally , Y is indeomp osable. This pro v es (v).  One no w denes left and righ t m utations. Corollary 3.15 b elo w laims that under some mild assumptions, these t w o notions oinide. Denition 3.8. Retaining the notation of prop osition 3.7 , one writes µ r X p add p T , k r Γ s b X qq  add p T , k r Γ s b Y q µ l X p add p T , k r Γ s b X qq  add p T , k r Γ s b Y 1 q . The map µ r X is alled the right X -mutation and µ l X is alled the left X -mutation . The ordered pair p X, Y q (resp. p X, Y 1 q ) is alled a right (r esp. left) exhange p air asso iate d with T . R emark 3.9 . This is not am biguous. Indeed, if T 1 P Add p C Γ q mod k r Γ s and X P T 1 is indeom- p osable then there exists a unique T P Add p T 1 q mod k r Γ s su h that add p k r Γ s b X q X T  0 and T 1  add p T , k r Γ s b X q : it is the full sub ategory of T 1 onsisting of ob jets whi h ha v e no ommon fators with k r Γ s b X . Lemma 3.10. L et p X, Y q b e a left (r esp. right) exhange p air. Then, (i) # X  # Y and ℓ p X q  ℓ p Y q ; (ii) if p X 1 , Y 1 q is another left (r esp. right) exhange p air and if X 1 P add p k r Γ s b X q , then Y 1 P add p k r Γ s b Y q . Pr o of. Let 0 Ñ X Ñ T Ñ Y Ñ 0 b e an admissible short exat sequene orresp onding to the ex hange pair. Using lemmas 2.36, 2.35 , 2.39 and the fat that the lass of admissi- ble monomorphisms (resp. epimorphisms) is stable under diret summands, one pro v es that 25 0 Ñ F X Ñ F T Ñ F Y Ñ 0 is a diret sum of admissible short exat sequenes of the same form as in prop osition 3.7 . Th us, ℓ p X q  ℓ p Y q . Moreo v er, it is lear that for t w o isomorphi indeom- p osable diret summands of F X , the t w o orresp onding indeomp osable diret summands of F Y are also isomorphi, and on v ersely . As a onsequene, # X  # Y . Let 0 Ñ X 1 Ñ T 1 Ñ Y 1 Ñ 0 b e an admissible short exat sequene orresp onding to the ex hange pair p X 1 , Y 1 q . If X 0 is an indeomp osable diret summand of F X , then, as add p k r Γ s b X q  add p X 0 r Γ sq  add p k r Γ s b X 1 q , X 0 is also an indeomp osable diret summand of F X 1 . Hene, the admissible short exat se- quene 0 Ñ X 0 f Ý Ñ T 0 Ñ Y 0 Ñ 0 where f is a minimal left add p F T 1 q -appro ximation app ears b oth as a diret summand of 0 Ñ F X Ñ F T Ñ F Y Ñ 0 and as a diret summand of 0 Ñ F X 1 Ñ F T 1 Ñ F Y 1 Ñ 0 . Finally F Y and F Y 1 ha v e Y 0 as a ommon diret summand, whi h implies that Y 1 P add p Y 0 r Γ sq  add p k r Γ s b Y q b eause Y and Y 1 are indeomp osable.  Lemma 3.11. L et X, Y P C Γ b e inde  omp osable, with ℓ p X q  ℓ p Y q and # X  # Y . The fol lowing ar e e quivalent: (i) F or every inde  omp osable obje t X 1 P add p k r Γ s b X q , dim Ext 1 C Γ p X 1 , Y q  " 1 if X 1  X 0 else. (ii) F or every inde  omp osable obje t Y 1 P add p k r Γ s b Y q , dim Ext 1 C Γ p X, Y 1 q  " 1 if Y 1  Y 0 else. Pr o of. Let r X (resp. r Y ) b e the set of isomorphism lasses of indeomp osable diret summands of k r Γ s b X (resp. k r Γ s b Y ). Using lemma 2.31 , dim Ext 1 C p F X , F Y q  dim Ext 1 C Γ p k r Γ s b X, Y q  ¸ X 1 P r X ℓ p X q ℓ p X 1 q # X 1 dim Ext 1 C Γ p X 1 , Y q (1)  dim Ext 1 C Γ p X, k r Γ s b Y q  ¸ Y 1 P r Y ℓ p Y q ℓ p Y 1 q # Y 1 dim Ext 1 C Γ p X, Y 1 q (2) If (i ) holds, then (1) yields dim Ext 1 C p F X , F Y q  ℓ p X q 2 { # X. As dim Ext 1 C Γ p X, Y q  1 , the orresp onding term in (2) is equal to ℓ p Y q 2 { # Y  ℓ p X q 2 { # X , and therefore all other terms v anish. The on v erse is pro v ed similarly .  Denition 3.12. With the notation of lemma 3.11 , if the t w o equiv alen t assumptions are satised, X and Y are alled neighb ours . Lemma 3.13. If X, Y P C Γ ar e inde  omp osable obje ts satisfying ℓ p X q  ℓ p Y q and # X  # Y , the fol lowing ar e e quivalent: (i) F or every inde  omp osable obje t X 0 P add p F X q , ther e exists an inde  omp osable obje t Y 0 P add p F Y q suh that for every inde  omp osable Y 1 0 P add p F Y q , dim Ext 1 C p X 0 , Y 1 0 q  " 1 if Y 1 0  Y 0 0 else. (ii) F or every inde  omp osable obje t Y 0 P add p F Y q , ther e exists an inde  omp osable obje t X 0 P add p F X q suh that for every inde  omp osable X 1 0 P add p F X q , dim Ext 1 C p X 1 0 , Y 0 q  " 1 if X 1 0  X 0 0 else. (iii) dim Ext 1 C p F X , F Y q  ℓ p X q 2 { # X . Mor e over, if X and Y ar e neighb ours then these thr e e  onditions ar e satise d. 26 Pr o of. By symmetry , it is enough to pro v e that (i) and (iii ) are equiv alen t. Let X 0 P add p F X q . Let F X  À n k  1 X i and F Y  À n k  1 Y i where the X i and the Y i are indeomp osable in C . Let r X b e a set of represen tativ es of isomorphism lasses of indeomp osable summands of F X and r Y a set of represen tativ es of isomorphism lasses of indeomp osable summands of F Y . dim Ext 1 C p F X , F Y q  n ¸ i  1 n ¸ j  1 dim Ext 1 C p X i , Y j q   ℓ p X q # X  2 ¸ X 1 0 P r X ¸ Y 1 0 P r Y dim Ext 1 C p X 1 0 , Y 1 0 q  ℓ p X q 2 # X ¸ Y 1 0 P r Y dim Ext 1 C p X 0 , Y 1 0 q whi h yields the equiv alene b eause the dim Ext 1 C p X 0 , Y 1 0 q are non-negativ e in tegers. The pro of of lemma 3.10 implies that if X and Y are neigh b ours, then (iii ) is satised.  Prop osition 3.14. L et X, Y P C Γ b e neighb ours suh that k r Γ s b X and k r Γ s b Y ar e rigid. L et 0 Ñ X f Ý Ñ M g Ý Ñ Y Ñ 0 b e a non-split admissible short exat se quen e (whih is unique up to isomorphism b e  ause X and Y ar e neighb ours). Then add p k r Γ s b p M ` X qq and add p k r Γ s b p M ` Y qq ar e rigid and X, Y R add p k r Γ s b M q . Mor e over, if ther e is T P Add p C Γ q mod k r Γ s suh that add p T , k r Γ s b X q and add p T , k r Γ s b Y q ar e maximal mo d k r Γ s -stable rigid, then f is a minimal left T -appr oximation and g is a minimal right T -appr oximation. Pr o of. In order to sho w that Ext 1 C Γ p k r Γ s b M , k r Γ s b X q  0 , it is enough to sho w that Ext 1 C Γ p M , X 1 q  0 for ev ery indeomp osable ob jet X 1 P add p k r Γ s b X q . Let X 1 P add p k r Γ s b X q b e indeomp osable. Applying Hom C Γ p , X 1 q to the admissible short exat sequene yields the follo wing long exat sequene: 0 Ñ Hom C Γ p Y , X 1 q Ñ Hom C Γ p M , X 1 q Hom C Γ p f ,X 1 q Ý Ý Ý Ý Ý Ý Ý Ý Ñ Hom C Γ p X, X 1 q δ Ý Ñ Ext 1 C Γ p Y , X 1 q Ñ Ext 1 C Γ p M , X 1 q Ñ Ext 1 C Γ p X, X 1 q  0 If X 1  X then dim Ext 1 C Γ p Y , X 1 q  1 . Th us, it is enough to pro v e that δ  0 , whi h is equiv alen t to Hom C Γ p f , X 1 q is not surjetiv e. If it w as surjetiv e, there w ould exist g P Hom C Γ p M , X 1 q su h that g f is an isomorphim. As a onsequene, rp g f q  1 g s f  Id X and the admissible short exat sequene w ould split, whi h is not the ase. If X 1 and X are not isomorphi, as X and Y are neigh b ours, Ext 1 C Γ p Y , X 1 q  0 and the result is lear. Similarly , Ext 1 C Γ p k r Γ s b M , k r Γ s b Y q  0 . In order to sho w that Ext 1 C Γ p k r Γ s b M , k r Γ s b M q  0 , it is enough to pro v e that Ext 1 C Γ p k r Γ s b M , M q  0 . Applying the funtor Hom C Γ p k r Γ s b M , q to the admissible short exat sequene indues the follo wing long exat sequene: Ext 1 C Γ p k r Γ s b M , X q  0 Ñ Ext 1 C Γ p k r Γ s b M , M q Ñ Ext 1 C Γ p k r Γ s b M , Y q  0 whi h yields the result. If X w as in add p k r Γ s b M q , there w ould exist an indeomp osable ob jet X 1 P add p k r Γ s b X q su h that X 1 P add p M q . Let M  X 1 ` M 1 . If X 1  X , then dim Ext 1 C Γ p Y , X q  1 and Ext 1 C Γ p Y , M q  0 are on traditory . Supp ose that X 1 and X are not isomorphi. Then there exists an indeomp osable ob jet Y 1 P add p k r Γ s b Y q su h that Ext 1 C Γ p X 1 , Y 1 q  0 and, b y denition of neigh b ours, Y 1 and Y are not isomorphi. Applying the funtor Hom C Γ p Y 1 , q to the admissible short exat sequene yields the long exat sequene Ext 1 C Γ p Y 1 , X q  0 Ñ Ext 1 C Γ p Y 1 , X 1 ` M 1 q Ñ Ext 1 C Γ p Y 1 , Y q  0 27 the rst equalit y oming from the fat that Y 1 and Y are not isomorphi and that X and Y are neigh b ours. As a onsequene, the en tral term v anishes, whi h on tradits the h yp othesis. Supp ose no w that T exists. F or T P T , applying Hom C Γ p T , q yields the follo wing long exat sequene Ext 1 C Γ p T , X q  0 Ñ Ext 1 C Γ p T , M q Ñ Ext 1 C Γ p T , Y q  0 and therefore, Ext 1 C Γ p T , M q  0 . Hene add p T , k r Γ s b p M ` X qqq is mo d k r Γ s -stable rigid. As a onsequene, as add p T , k r Γ s b X q is maximal mo d k r Γ s -stable rigid, M P add p T , k r Γ s b X q . As add p k r Γ s b X q X add p k r Γ s b M q  0 , one gets M P T . Applying Hom C Γ p , T q giv es rise to the follo wing long exat sequene: 0 Ñ Hom C Γ p Y , T q Ñ Hom C Γ p M , T q Hom C Γ p f ,T q Ý Ý Ý Ý Ý Ý Ý Ñ Hom C Γ p X, T q Ñ Ext 1 C Γ p Y , T q  0 and, as a onsequene, Hom C Γ p f , T q is surjetiv e for ev ery T P T , and the morphism f is a left T -appro ximation. If f w ere not minimal, there w ould exist a deomp osition M  M 0 ` M 1 with M 0  0 su h that f  f 0 ` f 1 where f 1  π M 1 f and f 0  0 . Th us, f 0 and f 1 w ould b e admissible monomorphisms. As Y is indeomp osable, one of the ok ernels of f 0 and f 1 v anishes. As cok er f 0  M 0  0 , cok er f 1  0 , and, as a onsequene, the admissible short exat sequene splits whi h is a on tradition. In the same w a y , g is a minimal righ t T -appro ximation.  Corollary 3.15. L et T P Add p C Γ q mod k r Γ s and X P C Γ b e an inde  omp osable obje t suh that X R T and add p T , k r Γ s b X q is maximal mo d k r Γ s -stable rigid. Then, the fol lowing ar e e quivalent: (i) Ther e exists an inde  omp osable obje t Y P C Γ suh that µ l X p add p T , k r Γ s b X qq  add p T , k r Γ s b Y q and X and Y ar e neighb ours. (ii) Ther e exists an inde  omp osable obje t Y 1 P C Γ suh that µ r X p add p T , k r Γ s b X qq  add p T , k r Γ s b Y 1 q and X and Y 1 ar e neighb ours. In this  ase, Y  Y 1 and if one denotes µ X p add p T , k r Γ s b X qq  µ l X p add p T , k r Γ s b X qq  µ r X p add p T , k r Γ s b X qq , then µ Y p µ X p add p T , k r Γ s b X qq  add p T , k r Γ s b X q . Pr o of. If (i) is true, prop osition 3.14 indues the admissible short exat sequene 0 Ñ X Ñ T 1 Ñ Y Ñ 0 satisfying the onditions of prop osition 3.7 , whi h pro v es (ii ) and the fat that Y  Y 1 . Hene µ Y p µ X p add p T , k r Γ s b X qq  µ l Y p µ r X p add p T , k r Γ s b X qq  add p T , k r Γ s b X q . By a similar argumen t, (ii) implies (i).  Denition 3.16. In the situation of the orollary , w e write µ X p add p T , k r Γ s b X qq  add p T , k r Γ s b Y q , and w e sa y that t X, Y u is an exhange p air asso iate d with T . 3.2. Rigid quasi-appro ximations. Denition 3.17. Let X P C Γ (resp. P C ). An epimorphism f : X ։ Y will b e alled a left rigid quasi-appr oximation of X if the follo wing onditions are satised:  k r Γ s b Y (resp. À g P Γ g b Y ) is rigid;  Y has an injetiv e en v elop e, without diret summand in add p k r Γ s b X q  resp. add  à g P Γ g b X  ;  If k r Γ s b Z (resp. À g P Γ g b Z ) is rigid, then ev ery morphism from X to Z , without in v ertible matrix o eien t, fatorizes through f . R emark 3.18 . The denition of a righ t rigid quasi-appro ximation is obtained from the previous one b y rev ersing the arro ws. All the follo wing results an b e adapted to this ase. 28 R emark 3.19 . As for the ase of appro ximations, minimal quasi-appro ximations are unique up to (non unique) isomorphism. The follo wing lemma giv es an easy w a y to get quasi-appro ximations: Lemma 3.20. L et D b e an ab elian  ate gory endowe d with an ation of Γ . L et C b e an exat ful l sub  ate gory of D . L et P b e a pr oje tive inde  omp osable obje t of C . If  every monomorphism of C to a pr oje tive obje t is admissible,  every monomorphism of C fr om P to an inde  omp osable obje t is admissible,  P has a simple so le S in D ,  À g P Γ g b S is rigid in D ,  the  okernel of S ã Ñ P is in C , then this  okernel is a left rigid quasi-appr oximation of P in C . Pr o of. Consider the follo wing short exat sequene in D : 0 Ñ S Ñ P Ñ Q Ñ 0 . Applying the funtor Hom D  À g P Γ g b S,   yields the follo wing long exat sequene: 0 Ñ Hom D  à g P Γ g b S, S  Ñ Hom D  à g P Γ g b S, P  Ñ Hom D  à g P Γ g b S, Q  Ñ Ext 1 D  à g P Γ g b S, S   0 . As the so le of P is simple, dim k Hom D  à g P Γ g b S, P   # t g P Γ | g b S  S u  dim k Hom D  à g P Γ g b S, S  and therefore Hom D  À g P Γ g b S, Q   0 . Applying the funtor Hom D   , À g P Γ g b Q  yields the long exat sequene 0 Ñ Hom D  Q, à g P Γ g b Q  Ñ Hom D  P , à g P Γ g b Q  Ñ Hom D  S, à g P Γ g b Q  Ñ Ext 1 D  Q, à g P Γ g b Q  Ñ Ext 1 D  P , à g P Γ g b Q   0 and, as Hom D  S, À g P Γ g b Q   0 , one gets Ext 1 D  Q, à g P Γ g b Q   0 . Let no w S 1 b e the (not neessarily simple) so le of Q . As Hom D  à g P Γ g b S, Q   0 , S 1 do es not on tain an y diret summand isomorphi to some g b S . Hene, if one denotes α : S 1 Ñ I the injetiv e en v elop e of S 1 , I has no diret summand of the form g b P . Moreo v er, 29 as I is injetiv e, one an omplete the follo wing omm utativ e diagram: S 1 α / / ι   I Q β ? ?         Here, β is an admissible monomorphism sine I is injetiv e. Hene, the seond ondition of the denition of a left quasi-appro ximation is pro v ed. Let no w Z P C b e indeomp osable and f : P Ñ Z . Supp ose that f is not a monomorphism. Let K Ñ P b e the k ernel of f in D and let S 1 b e the so le of K . By omp osing the t w o morphisms, one has a non v anishing morphism g : S 1 Ñ P su h that f g  0 . As the so le of P is simple, S 1  S and nally , f v anishes on S and therefore fatorizes through P Ñ Q whi h is the ok ernel of S Ñ P . If f is a monomorphism, b y h yp othesis, it is admissible in C so it splits and nally f  Id P as Z is indeomp osable. The third ondition is pro v ed.  Lemma 3.21. If every inde  omp osable pr oje tive obje t of C has a left rigid quasi-appr oximation, then every inde  omp osable pr oje tive obje t of C Γ has a left rigid quasi-appr oximation. Pr o of. Let p P , ψ q P C Γ b e an indeomp osable pro jetiv e ob jet. By lemma 2.44 , P P C is pro jetiv e. Let f : P ։ X b e the diret sum of the minimal left rigid quasi-appro ximations of its indeomp osable diret summands. It is also a minimal left rigid quasi-appro ximation sine add X  add  à g P Γ g b X 0  where P 0 ։ X 0 is one of the minimal left rigid quasi-appro ximations of the indeomp osable diret summands of P . Hene À g P Γ g b X is rigid. Therefore, for all g P Γ , there exists a unique morphism χ g whi h mak es the follo wing diagram omm utativ e: g b P g b f / / ψ g   g b X χ g   P f / / X (The existene omes from the denition of a rigid quasi-appro ximation, the uniit y omes from the fat that f is an epimorphism). Clearly , p X, χ q P C Γ . So it is easy to see that f : p P , ψ q ։ p X, χ q is a left rigid quasi-appro ximation.  3.3. Endomorphisms. All pro jetiv e indeomp osable ob jets of C will b e supp osed here to ha v e left rigid quasi-appro ximations. All results remain v alid if they are supp osed to ha v e righ t rigid quasi-appro ximations. Let T P add p C Γ q mod k r Γ s b e maximal mo d k r Γ s -stable rigid. Let T P C Γ and r T P F T b e basi su h that T  add p T q and F T  add p r T q (one an nd su h T and r T sine T is nitely generated). W rite E  End C Γ p T q and r E  End C p r T q . If X P T is indeomp osable, denote b y S X the orresp onding simple represen tation of E , that is, the head of the pro jetiv e E -mo dule Hom C p X, T q . Lik ewise, if X P F T is indeomp osable, denote b y r S X the orresp onding simple represen tation of r E . Denition 3.22. Let D P Add p C Γ q mod k r Γ s and let X P D b e indeomp osable. A mo d k r Γ s -lo op of D at X is an irreduible morphism X Ñ X 1 of D where X 1 P add p k r Γ s b X q is indeomp osable. A mo d k r Γ s - 2 -yle of D at X is a ouple of irreduible morphisms X Ñ Y and Y Ñ X 1 of D where X 1 P add p k r Γ s b X q is indeomp osable. Denition 3.23. Let D P Add p C q Γ and X P D b e indeomp osable. A Γ -lo op of D at X is an irreduible morphism X Ñ g b X of D where g P Γ . A Γ - 2 -yle of D at X is a ouple of irreduible morphisms X Ñ Y and Y Ñ g b X of D where g P Γ . 30 Lemma 3.24. L et D P Add p C Γ q mod k r Γ s . L et X P D b e inde  omp osable and X 1 b e a dir e t summand of F X . Then D has no mo d k r Γ s -lo ops (r esp. mo d k r Γ s - 2 -yles) at X if and only if F D has no Γ -lo ops (r esp. Γ - 2 -yles) at X 1 . Pr o of. The pro of is the same for lo ops and 2 -yles. Hene, it will b e done only for lo ops. Supp ose that D has no mo d k r Γ s -lo ops at X . Let f P End C p F X q . Then f r Γ s P End C Γ pp F X qr Γ sq  End C Γ p k r Γ s b X q . As D has no mo d k r Γ s -lo ops at X , f r Γ s fatorizes through Y P D su h that add p Y q X add p k r Γ s b X q  0 . Hene F p f r Γ sq fatorizes through F Y . As add p k r Γ s b X q is mo d k r Γ s -stable, add p F Y q X add p F p k r Γ s b X qq  0 . As f is a diret summand of F p f r Γ sq , f fatorizes also through F Y and, as a onsequene, F D has no Γ -lo ops at X 1 . Con v ersely , supp ose that F D has no Γ -lo ops at X 1 . Let f P End C Γ p X q . As F f P End C p F X q , F f fatorizes through Y P D su h that add p Y q X add  À g P Γ g b X 1   0 . Th us, p F f qr Γ s fatorizes through Y r Γ s and f , as a diret summand of p F f qr Γ s , fatorizes also through Y r Γ s and nally , as add  à g P Γ g b X 1   add F p X 1 r Γ sq  add F X, one gets add p Y r Γ sq X add p X q  0 . Ev en tually , D has no mo d k r Γ s -lo ops at X .  Lemma 3.25. L et T 0 P Add p T q mod k r Γ s and let p X, Y q b e a left (r esp. right) exhange p air asso iate d to T 0 suh that add p T 0 , k r Γ s b X q  T . The fol lowing ar e e quivalent: (i) T has no mo d k r Γ s -lo ops at X ; (ii) F or al l inde  omp osable X 1 P add p k r Γ s b X q , every non invertible morphism fr om X to X 1 fatorizes thr ough T 0 . (iii) X and Y ar e neighb ours. Pr o of. F or the pro of, the ex hange pair will b e onsidered to b e a left ex hange pair. The equiv alene of (i) and (ii ) is lear. Let 0 Ñ X f Ý Ñ T 1 Ñ Y Ñ 0 b e the admissible short exat sequene orresp onding to the ex hange pair p X, Y q . Let X 1 P add p k r Γ s b X q . Applying Hom C Γ p , X 1 q leads to the long exat sequene: 0 Ñ Hom C Γ p Y , X 1 q Ñ Hom C Γ p T 1 , X 1 q Hom C Γ p f ,X 1 q Ý Ý Ý Ý Ý Ý Ý Ý Ñ Hom C Γ p X, X 1 q Ñ Ext 1 C Γ p Y , X 1 q Ñ Ext 1 C Γ p T 1 , X 1 q  0 . As f is a left T 0 -appro ximation, for an y elemen t of Hom C Γ p X, X 1 q , fatorizing through T 0 is equiv alen t to b eing in the image of Hom C Γ p f , X 1 q . There are t w o ases:  if X 1  X then ev ery non in v ertible morphism from X to X 1 fatorizes through T 0 if and only if Ext 1 C Γ p Y , X 1 q  k b eause, as k is algebraially losed, End C Γ p X q{ m  k where m is the maximal ideal of End C Γ p X q ;  if X 1 and X are not isomorphi then ev ery morphism from X to X 1 fatorizes through T 0 if and only if Ext 1 C Γ p Y , X 1 q  0 . Com bining these t w o ases, the equiv alene of ( ii) and (iii ) is pro v ed.  Prop osition 3.26. Supp ose that T has no mo d k r Γ s -lo ops. Then gl . dim p E q  gl . dim p r E q " 6 2 if T is pr oje tive  3 else. Pr o of. The pro of is the same for E and for r E sine r T has no Γ -lo ops and sine T is pro jetiv e if and only if r T is pro jetiv e. Hene, it is enough to do it for E . Let X P T b e indeomp osable and T 0 P T mod k r Γ s b e su h that T  add p T 0 , k r Γ s b X q and X R T 0 . Supp ose that X is not pro jetiv e. There exist t w o ex hange admissible short exat sequenes 0 Ñ X f Ý Ñ T 1 Ñ Y Ñ 0 and 0 Ñ Y Ñ T 2 Ñ X Ñ 0 31 sine, using the previous lemma, X and Y are neigh b ours and using orollary 3.15 , one gets µ Y p µ X p T qq  T . Applying Hom C Γ p , T q to these sequenes yields the follo wing long exat sequenes: 0 Ñ Hom C Γ p Y , T q Ñ Hom C Γ p T 1 , T q Ñ Hom C Γ p X, T q Ñ Ext 1 C Γ p Y , T q  Ext 1 C Γ p Y , X 1 q Ñ Ext 1 C Γ p T 1 , T q  0 where X 1 is the largest diret summand of T on tained in add p k r Γ s b X q and 0 Ñ Hom C Γ p X, T q Ñ Hom C Γ p T 2 , T q Ñ Hom C Γ p Y , T q Ñ Ext 1 C Γ p X, T q  0 As T is basi and X 1 is mo d k r Γ s -stable, X 1 on tains exatly one ob jet of ea h isomorphism lass of add p k r Γ s b X q . As X and Y are neigh b ours, dim Ext 1 C Γ p Y , X 1 q  1 . Th us, Ext 1 C Γ p Y , X 1 q  S X as an E -mo dule. Therefore, om bining these t w o long exat sequenes yields the follo wing long exat sequene of E -mo dules: 0 Ñ Hom C Γ p X, T q Ñ Hom C Γ p T 2 , T q Ñ Hom C Γ p T 1 , T q Ñ Hom C Γ p X, T q Ñ S X Ñ 0 whi h is a pro jetiv e resolution of S X . As a onsequene, pro j . dim p S X q 6 3 . As X R add p T 2 q , Hom E p Hom C Γ p T 2 , T q , S X q  0 and therefore Ext 3 p S X , S X q  Hom E p Hom C Γ p X, T q , S X q has dimension 1 . Finally , pro j . dim p S X q  3 . Supp ose no w that X is pro jetiv e. Sho w that pro j . dim p S X q 6 2 . Let π : X ։ Y b e a left rigid quasi-appro ximation. By denition, the injetiv e en v elop e of Y do es not in terset add p k r Γ s b X q and, using lemma 2.38 , there is an admissible short exat sequene 0 Ñ Y f Ý Ñ T 1 Ñ Z Ñ 0 where f is a left add p T 0 q -appro ximation. Using lemma 3.6 and as X is pro jetiv e, add p T , k r Γ s b Z q is mo d k r Γ s -stable rigid, and, sine T is maximal mo d k r Γ s -stable rigid, Z P T . Applying Hom C Γ p , T q to this admissible short exat sequene yields the long exat sequene 0 Ñ Hom C Γ p Z, T q Ñ Hom C Γ p T 1 , T q Ñ Hom C Γ p Y , T q Ñ Ext 1 C Γ p Z, T q  0 Moreo v er, Hom C Γ p π , T q : Hom C Γ p Y , T q Ñ Hom C Γ p X, T q is injetiv e b eause π is an epimor- phism, and its ok ernel has dimension 1 : this ok ernel is S X . One dedues the follo wing long exat sequene: 0 Ñ Hom C Γ p Z, T q Ñ Hom C Γ p T 1 , T q Ñ Hom C Γ p X, T q Ñ S X Ñ 0 whi h is a pro jetiv e resolution of S X .  Reall this theorem of Happ el: Theorem 3.27 ([Hap , setion 1.4℄) . If A is a k -algebr a and X is a tilting A -mo dule, then mo d A and mo d End A p X q op ar e derive d e quivalent. The follo wing prop osition explains the relationship b et w een an y maximal mo d k r Γ s -stable rigid ategory T 1 and the initial one T . Prop osition 3.28. Supp ose that T 1 P Add p C Γ q mod k r Γ s is maximal mo d k r Γ s -stable rigid. Then T 1 is nitely gener ate d. L et T 1 P T 1 b e b asi suh that T 1  add p T 1 q . L et E 1  End C Γ p T 1 q . L et M  Hom C Γ p T 1 , T q . Then M is a tilting mo dule on E and End E p M q  E 1 op . In p artiular, ther e is a derive d e quivalen e b etwe en E and E 1 and T and T 1  ontain the same numb er of inde  omp osable obje ts up to isomorphism. The same holds for r T 1 and r E 1 . 32 Pr o of. Using lemma 2.38 , one gets the admissible short exat sequene (3) 0 Ñ T 1 2 Ñ T 1 1 g Ý Ñ T Ñ 0 where g is a minimal righ t T 1 -appro ximation. Using lemma 3.6, T 1 2 P T 1 . Let T 1 P add p T 1 q b e basi su h that T 1 1 , T 1 2 P add p T 1 q (in fat add p T 1 q  T 1 will b e pro v ed later). By the same argumen t, there is an admissible short exat sequene (4) 0 Ñ T 1 f Ý Ñ T 1 Ñ T 2 Ñ 0 where f is a minimal left T -appro ximation and T 2 P T . Applying Hom C Γ p , T q to (4 ) yields the follo wing long exat sequene: (5) 0 Ñ Hom C Γ p T 2 , T q Ñ Hom C Γ p T 1 , T q Ñ Hom C Γ p T 1 , T q  M Ñ Ext 1 C Γ p T 2 , T q  0 and, as a onsequene, pro j . dim E M 6 1 . No w, applying the funtor Hom C Γ p T 1 , q to (4 ) giv es the follo wing long exat sequene: 0 Ñ Hom C Γ p T 1 , T 1 q Ñ Hom C Γ p T 1 , T 1 q Ñ Hom C Γ p T 1 , T 2 q Ñ Ext 1 C Γ p T 1 , T 1 q  0 . Applying the funtor Hom E p , M q to (5) indues the long exat sequene 0 Ñ Hom E p Hom C Γ p T 1 , T q , M q Ñ Hom E p Hom C Γ p T 1 , T q , M q Ñ Hom E p Hom C Γ p T 2 , T q , M q Ñ Ext 1 E p Hom C Γ p T 1 , T q , M q Ñ Ext 1 E p Hom C Γ p T 1 , T q , M q  0 the last equalit y oming from the fat that Hom C Γ p T 1 , T q is a pro jetiv e E -mo dule. Let us sho w that the morphism of funtors from T to mo d E 1 op Φ : Hom C Γ p T 1 , q Ñ Hom E p Hom C Γ p , T q , Hom C Γ p T 1 , T qq ϕ ÞÑ Hom C Γ p ϕ, T q is an isomorphism. By additivit y and sine T  add p T q , it is enough to lo ok at Φ T . Let ϕ P Hom C Γ p T 1 , T q . Then Φ T p ϕ qp Id T q  ϕ . Finally , Φ T is injetiv e. Moreo v er, if ϕ 1 P Hom E p Hom C Γ p T , T q , Hom C Γ p T 1 , T qq then Φ T p ϕ 1 p Id T qq  ϕ 1 so that Φ T is surjetiv e. By omparing the t w o previous long exat sequenes, one gets the follo wing isomorphisms of E 1 op -mo dules. End E p M q  Hom E p Hom C Γ p T 1 , T q , M q  End C Γ p T 1 q  E 1 op and Ext 1 E p M , M q  Ext 1 E p Hom C Γ p T 1 , T q , M q  0 . Applying Hom C Γ p , T q to (3) yields the follo wing long exat sequene: 0 Ñ Hom C Γ p T , T q Ñ Hom C Γ p T 1 1 , T q Ñ Hom C Γ p T 1 2 , T q Ñ Ext 1 C Γ p T , T q  0 . Hene, M is a tilting E -mo dule. F rom theorem 3.27 , one dedues that E and E 1 are deriv ed equiv alen t and that T and add p T 1 q on tain the same n um b er of isomorphism lasses of indeom- p osable ob jets. It is no w ob vious that add p T 1 q  T 1 (if X P T 1 z add p T 1 q , the pro of an b e done replaing T 1 b y T 1 ` X and w e w ould obtain that T , add p T 1 ` X q and add p T 1 q on tain the same n um b er of isomorphism lasses of indeomp osable ob jets. This w ould b e a on tradition).  Reall this theorem of Igusa: Theorem 3.29 ([Igu , 3.2, b℄) . L et A b e a k -algebr a of nite dimension and nite glob al dimen- sion. L et ϕ b e an automorphism of A suh that ther e exists a family of primitive idemp otents of A on whih ϕ ats as a p ermutation. Then, the Gabriel quiver of A has no arr ows b etwe en any two verti es of the same orbit of ϕ . Reall also the follo wing theorem of Iy ama, in a partiular ase: Theorem 3.30 ([Iy a1, 5.1 (3)℄) . Assume that r T 1 P Add p C q is rigid and  ontains the pr oje tive- inje tive obje ts of C . L et r T 1 P r T 1 b e b asi suh that r T 1  add p r T 1 q , r E 1  End C p r T 1 q . Then gl . dim p r E 1 q 6 3 if and only if r T 1 is luster-tilting. 33 W e shall also need the follo wing prop osition of Bongartz: Prop osition 3.31 ([Bon , p. 463℄) . L et Q b e a quiver and I b e an admissible ide al of k Q suh that k Q { I is nite dimensional. Denote by J the Ja obson r adi al of k Q (that is the ide al of k Q gener ate d by arr ows). L et i, j P Q 0 . Then dim e j p I {p I J  J I qq e i  dim Ext 2 mod kQ { I p S i , S j q wher e S i and S j ar e the simple r epr esentations of k Q supp orte d on verti es i and j . Finally , reall a partiular ase of a theorem b y Lenzing: Theorem 3.32 ([Len , satz 5℄) . If A is a k -algebr a of nite dimension and nite glob al dimension, then every nilp otent element of A is in the additive sub gr oup r A, A s gener ate d by  ommutators. W e an no w state and pro v e the main result of this setion. Theorem 3.33. Supp ose that ther e exists a  ate gory T P add p C Γ q mod k r Γ s whih is maximal mo d k r Γ s -stable rigid without mo d k r Γ s -lo ops. L et T 1 P add p C Γ q mod k r Γ s b e maximal mo d k r Γ s -stable rigid, r T 1  F T 1 . L et T 1 P T 1 , r T 1 P r T 1 b e b asi suh that T 1  add p T 1 q and r T 1  add p r T 1 q . L et E 1  End C Γ p T 1 q and r E 1  End C p r T 1 q . Then: (i) T 1 has no mo d k r Γ s -lo ops; r T 1 has no Γ -lo ops; (ii) gl . dim p E 1 q  gl . dim p r E 1 q "  3 if T 1 is not pr oje tive 6 2 else; (iii) T 1 and r T 1 ar e luster-tilting; (iv) for every simple E 1 -mo dules S and S 1 suh that add p k r Γ s b S q  add p k r Γ s b S 1 q , one has Ext 1 E 1 p S, S 1 q  Ext 2 E 1 p S, S 1 q  0 ; for every simple r E 1 -mo dules S and S 1 suh that add  À g P Γ g b S   add  À g P Γ g b S 1  , one has Ext 1 r E 1 p S, S 1 q  Ext 2 r E 1 p S, S 1 q  0 ; (v) T 1 has no mo d k r Γ s - 2 -yles; r T 1 has no Γ - 2 -yles. Pr o of. (i) Using lemma 3.24 , it is enough to pro v e that r T 1 has no Γ -lo ops. F or g P Γ , w e sho w that r T 1 has no x g y -lo ops. F or that, w e sho w that r E 1 satises the h yp othesis of theorem 3.29 . Using prop osition 3.28 , r E 1 is of nite global dimension sine r E is. F rom prop osition 2.7 , there exists X P C x g y su h that F X  r T 1 . It indues an ation of x g y on r E 1  Hom C x g y p X, X q . Using prop ositions 2.7 and 2.11 implies that F X an b e split up in to a diret summand of indeomp osable ob jets su h that x g y ats on it b y p erm uting these ob jets. Then x g y ats on the family of primitiv e idemp oten ts orresp onding to these ob jets b y p erm utation and therefore theorem 3.29 applies. (ii) This follo ws from (i ) and prop osition 3.26. (iii) This follo ws from (ii ) and theorem 3.30 . (iv) As T 1 (resp. r T 1 ) has no mo d k r Γ s -lo ops (resp. Γ -lo ops), Ext 1 E 1 p S, S 1 q  0 p resp. Ext 1 r E 1 p S, S 1 q  0 q . Conerning Ext 2 , if X P T 1 is not pro jetiv e, S X has the follo wing pro jetiv e resolution, giv en in prop osition 3.26 : 0 Ñ Hom C Γ p X, T q Ñ Hom C Γ p T 2 , T q Ñ Hom C Γ p T 1 , T q Ñ Hom C Γ p X, T q Ñ S X Ñ 0 . As add p T 2 q X add p k r Γ s b X q  0 , Hom E 1 p Hom C Γ p T 2 , T q , S X 1 q  0 and therefore Ext 2 E 1 p S X , S X 1 q  0 for ev ery indeomp osable ob jet X 1 P add p k r Γ s b X q . If X is pro jetiv e, the pro jetiv e resolution is 0 Ñ Hom C Γ p Z, T q Ñ Hom C Γ p T 1 , T q Ñ Hom C Γ p X, T q Ñ S X Ñ 0 . 34 As add p k r Γ s b X q X add p T 1 q  0 , add p k r Γ s b X q X add p Z q  0 and b y the same argumen t as b efore, Ext 2 E 1 p S X , S X 1 q  0 for an y indeomp osable ob jet X 1 P add p k r Γ s b X q . The same argumen t w orks for r E 1 . (v) It is enough to sho w this for T 1 , thanks to lemma 3.24 . Supp ose that T 1 has a mo d k r Γ s - 2 -yle. As End C Γ p T 1 q  Hom mod k r Γ s p 1 , Hom C Γ p T 1 , T 1 qq , there exist t w o arro ws r a and r b in the Gabriel quiv er of r T 1 su h that r a r b is a Γ - 2 -yle of r T 1 and ab is a mo d k r Γ s - 2 -yle of T 1 with a  ° g P Γ g  r a and b  ° g P Γ g  r b . As ab is nilp oten t, theorem 3.32 indues the follo wing iden tit y in E 1 : ab  n ¸ i  1 λ i r u i , v i s where, for ea h i , u i  ° g P Γ g  r u i and v i  ° g P Γ g  r v i where r u i and r v i are paths of the Gabriel quiv er of r T 1 . One an supp ose that r u 1  r a and r v 1  r b and, without loss of generalit y that r a r b and r b r a do not app ear as terms of r r u i , r v i s for i > 2 . If λ 1  1 , r a r b  e t p r a q r a r be s p r b q  1 1  λ 1   λ 1 e t p r a q r b r ae s p r b q  n ¸ i  2 e t p r a q λ i r u i , v i s e s p r b q  is a non trivial iden tit y in End C p F T 1 q (that is, an iden tit y whi h is false in the path algebra of the quiv er). It on tains r a r b with a non zero o eien t in the left hand side. If λ 1  1 , as r a r b is a Γ - 2 -yle, there exists h P Γ su h that t p r a q  h  s p r b q and e h  t p r b q p h  r b q r ae s p r a q   n ¸ i  2 λ i e h  t p r b q r u i , v i s e s p r a q is a non trivial iden tit y in End C p F T 1 q . It on tains p h  r b q r a with a non zero o eien t in the left hand side. In these t w o ases, one gets a non trivial relation, and as a onsequene, using prop o- sition 3.31 , Ext 2 C p S s p r b q , S t p r a q q  0 or Ext 2 C p S s p r a q , S t p h  r b q q  0 whi h on tradits (iv).  Denition 3.34. Let X, Y P C Γ . W e write X  Y if add p k r Γ s b X q  add p k r Γ s b Y q and X and Y are said to b e e quivalent mo dulo mo d k r Γ s . The follo wing theorem summarizes the results onerning m utation: Theorem 3.35. Supp ose that ther e exists a  ate gory T P Add p C Γ q maximal mo d k r Γ s -stable rigid whih has no mo d k r Γ s -lo ops. L et T 1 P Add p C Γ q b e maximal mo d k r Γ s -stable rigid. Then T 1 is luster-tilting (hen e maximal rigid). L et X P T 1 b e an inde  omp osable obje t of C Γ , and T 1 0 P Add p C Γ q mod k r Γ s satisfying X R T 1 0 and add p T 1 0 , k r Γ s b X q  T 1 . If X is pr oje tive, every Y P C Γ whih is inde  omp osable suh that add p T 1 0 , k r Γ s b Y q is maximal rigid is e quivalent to X mo dulo mo d k r Γ s . If X is not pr oje tive, ther e exists a unique Y P C Γ suh that t X, Y u is an exhange p air asso iate d with T 1 0 . Mor e over, in this  ase, X and Y ar e neighb ours. If X 1 P T 1 is e quivalent to X mo dulo mo d k r Γ s and if t X, Y u , t X 1 , Y 1 u ar e two exhange p airs asso iate d with T 1 0 , then Y and Y 1 ar e e quivalent mo dulo mo d k r Γ s . If X and Y denote the e quivalen e lasses of X and Y mo dulo mo d k r Γ s , one wil l denote µ X p T 1 q  add p T 1 0 , k r Γ s b Y q . Hen e one has µ X p µ Y p T 1 qq  T 1 . Pr o of. This follo ws from prop osition 3.7 , lemmas 3.25 and 3.10 , orollary 3.15 and theorem 3.33 . Note that if X is not pro jetiv e, the existene and uniit y of Y su h that p X, Y q is a left ex hange pair asso iated with T 1 0 is lear b y prop osition 3.7. Using lemma 3.25, one dedues that X and Y are neigh b ours. Therefore, orollary 3.15 implies that t X, Y u is an (unordered) ex hange pair.  35 3.4. Ex hange matries. As in the previous setion, indeomp osable pro jetiv e ob jets of C are supp osed to ha v e left rigid quasi-appro ximations. As b efore, all results remain v alid if they ha v e righ t rigid quasi-appro ximations. As b efore, T P add p C Γ q mod k r Γ s is maximal rigid mo d k r Γ s -stable (and nitely generated). One supp oses moreo v er that T has no mo d k r Γ s -lo ops. Let r T  F T . Thanks to the previous setion, T and r T are luster-tilting. Let T P C Γ and r T P C b e basi su h that T  add p T q and r T  add p r T q . One denotes b y Q the Gabriel quiv er of End C Γ p T q and b y r Q the quiv er of End C p r T q . Denote b y Q 0 { mo d k r Γ s the set of equiv alene lasses mo dulo mo d k r Γ s as in denition 3.34. There is a anonial bijetion b et w een the sets Q 0 { mo d k r Γ s and r Q 0 { Γ . If X P r Q 0 { Γ , X  P Q 0 { mo d k r Γ s will denote the image of X b y this bijetion. Denote b y P  Q 0 and r P  r Q 0 the sets of v erties orresp onding to pro jetiv e ob jets. Denition 3.36. If X P p r Q 0 z r P q{ Γ and Z P r Q 0 { Γ , put b Z X  # t q P r Q 1 | s p q q P Z , t p q q P X u  # t q P r Q 1 | s p q q P X , t p q q P Z u # X Denote b y B p T q the matrix ha ving these en tries. It will b e alled the exhange matrix of T . R emarks 3.37 .  As r T has no Γ - 2 -yles, # t q P r Q 1 | s p q q P X , t p q q P Z u  0 or # t q P r Q 1 | s p q q P Z , t p q q P X u  0 .  As r T has no Γ -lo ops, b X X  0 .  It is easy to see that b Z X  # t q P r Q 1 | s p q q P Z , t p q q  X u  # t q P r Q 1 | s p q q  X, t p q q P Z u for ev ery X P X , hene B p T q has in teger o eien ts.  The ex hange matrix is learly sk ew-symmetrizable (righ t m ultipliation b y the diagonal matrix p # X q X P p r Q 0 z r P q { Γ giv es a sk ew-symmetri matrix). R emark 3.38 . The matrix of denition 3.36 oinides with the ex hange matrix of [FK℄ and [GLS3 ℄. Fix no w the three follo wing m utations and matries:  µ on C Γ and B p T q are dened as b efore;  r µ is the m utation on C  C t e u dened as b efore replaing Γ b y the trivial group t e u and r B p T q is the orresp onding ex hange matrix;  µ  is the m utation on C Γ  p C Γ qt e u dened as b efore replaing C b y C Γ and Γ b y t e u and B  p T q is the orresp onding ex hange matrix. R emark 3.39 . The denitions of r µ and µ  oinide with those of [FK ℄ and [GLS3 ℄. Prop osition 3.40. L et X P p r Q 0 z r P q{ Γ and Z P r Q 0 { Γ . Then (i) F µ X p T q    ¹ X 1 P X r µ X 1   p r T q wher e the r µ X 1  ommute. (ii) F or every X P X , b Z X  ¸ Z P Z r b Z X . (iii) µ X p T q    ¹ X 1 P X  µ  X 1   p T q wher e the µ  X 1  ommute. (iv) F or every Z P Z  , b Z X  # Z ℓ p Z q ¸ X P X  ℓ p X q # X b  Z X . 36 Pr o of. (i) Let 0 Ñ X Ñ T Ñ Y Ñ 0 b e the admissible short exat sequene of C Γ orresp onding to the m utation µ X in T . Then X and Y are neigh b ours and, as a onsequene, one an write X  t X i | i P J 1 , # X K u and Y  t Y i | i P J 1 , # Y K u in su h a w a y that for ev ery i, j P J 1 , ℓ p X q K  J 1 , ℓ p Y q K , dim Ext 1 C p X i , Y i q  δ ij b y using lemma 3.13. Th us, for i P J 1 , ℓ p X q K , there exists a non split admissible short exat sequene 0 Ñ X i Ñ T i Ñ Y i Ñ 0 in C . As r T has no Γ -lo ops, none of the X j and Y j is in add p T i q and nally , the result is lear. (ii) This is an easy onsequene of the denition. (iii) The pro of is the same as for (i). (iv) Let X P X and T 0 P Add p C Γ q mod k r Γ s b e su h that add p X r Γ sq X F T 0  0 and T  add p T 0 , X r Γ sq . Let 0 Ñ X Ñ T Ñ Y Ñ 0 b e an admissible short exat sequene in C orresp onding to the m utation r µ X . As r T has no Γ -lo ops, T P add p F T 0 q . As a onsequene, lemma 2.39 giv es 0 Ñ X r Γ s Ñ T r Γ s Ñ Y r Γ s Ñ 0  à X 1 P X   0 Ñ X 1 Ñ T 1 X 1 Ñ Y 1 X 1 Ñ 0  ℓ p X 1 q{ # X 1 where, for ev ery X 1 P X  , 0 Ñ X 1 Ñ T 1 X 1 Ñ Y 1 X 1 Ñ 0 is an admissible short exat sequene orresp onding to µ  X 1 at T , and the exp onen ts ℓ p X 1 q{ # X 1 ome from lemma 2.31. Let no w Z P Z  . By denition of B  p T q , the n um b er of opies of Z in the middle term of the righ t hand side is ¸ X 1 P X  ℓ p X 1 q # X 1 max p 0 ,  b  Z X 1 q  max   0 ,  ¸ X 1 P X  ℓ p X 1 q # X 1 b  Z X 1   the equalit y oming from the fat that all b  Z X ha v e the same sign, b eause T has no mo d k r Γ s - 2 -yles. Moreo v er, the n um b er of opies of Z in the middle term of the left hand side is ℓ p Z q{ # Z times the n um b er of opies of an elemen t of Z in T , that is ℓ p Z q # Z ¸ Z 1 P Z max p 0 ,  r b Z 1 X q  max   0 ,  ℓ p Z q # Z ¸ Z 1 P Z r b Z 1 X    max  0 ,  ℓ p Z q # Z b Z X  . One dedues the iden tit y: max   0 ,  ¸ X 1 P X  ℓ p X 1 q # X 1 b  Z X 1    max  0 ,  ℓ p Z q # Z b Z X  and using the same argumen t for the admissible short exat sequene orresp onding to the m utation from Y to X , one dedues that max   0 , ¸ X 1 P X  ℓ p X 1 q # X 1 b  Z X 1    max  0 , ℓ p Z q # Z b Z X  . These t w o equalities yield the result.  R emark 3.41 . The form ula (ii ) w as giv en in Dynkin ases b y Dup on t in [ Dup2℄ (see also erratum [Dup1 ℄). On the other hand, (iv) is a generalization of the form ula giv en b y Y ang in [Y an ℄ for luster algebras of nite t yp e (for a yli group). Theorem 3.42. L et X P T b e an inde  omp osable non pr oje tive obje t in C Γ . Then B p µ X p T qq  µ X  p B p T qq wher e X is the orbit of X in Q 0 { mo d k r Γ s and µ X the mutation of matri es dene d by F omin and Zelevinsky. 37 Pr o of. The result is kno wn for r B p T q and r µ (the pro of is similar as the one in [GLS3 , 14℄ for example). Let X  t X 1 , X 2 , . . . X n u . F or i P J 1 , n K , denote b y i r b Z Y the o eien ts of the matrix r B  r µ X i r µ X i  1 . . . r µ X 1 p r T q  . Then, b y an easy indution, b y using the fat that r T has no Γ -lo ops nor Γ - 2 -yles, one has  i r b Z Y   r b Z Y if Z  X j or Y  X j where 1 6 j 6 i ;  i r b Z Y  r b Z Y if Z  X j or Y  X j where i  j 6 n ;  i r b Z Y  r b Z Y  i ¸ j  1 | r b Z X j | r b X j Y  r b Z X j | r b X j Y | 2 else. Denoting b y  b the o eien ts of B p µ X p T qq , b y using prop osition 3.40 , for Y P p r Q 0 z r P q{ Γ and Z P r Q 0 { Γ , for Y P Y ,   b Z Y  ¸ Z P Z n r b Z Y  ¸ Z P Z p r b Z Y q   b Z Y if Z  X or Y  X ;   b Z Y  ¸ Z P Z n r b Z Y  ¸ Z P Z  r b Z Y  n ¸ j  1 | r b Z X j | r b X j Y  r b Z X j | r b X j Y | 2  else. Then, using the fat that all r b Z X j (resp. all r b X j Y ) are of the same sign (as r T has no Γ - 2 -yles), ¸ Z P Z n ¸ j  1 | r b Z X j | r b X j Y  r b Z X j | r b X j Y | 2  n ¸ j  1    ° Z P Z r b Z X j    r b X j Y   ° Z P Z r b Z X j  | r b X j Y | 2  n ¸ j  1 | b Z X | r b X j Y  b Z X | r b X j Y | 2  | b Z X |  ° n j  1 r b X j Y   b Z X    ° n j  1 r b X j Y    2  | b Z X | b X Y  b Z X | b X Y | 2 and the result is pro v ed.  3.5. Cluster  haraters. The ideas of this setion generalize results of [FK℄. W e retain notation and h yp othesis of previous setions. One supp oses moreo v er that there exists T P Add p C q Γ maximal Γ -stable rigid without Γ -lo ops. Th us, T is luster-tilting using theorem 3.35 . One denotes b y T 1 , T 2 , . . . , T n the indeomp osable ob jets of T up to isomorphism, the T i for i P J r  1 , n K b eing the pro jetiv e ob jets. The ation of Γ on T indues an ation on J 1 , n K . If i P J 1 , n K , i denotes its equiv alene lass mo dulo Γ . One denotes b y I the set of these equiv alene lasses. Let T  T 1 ` T 2 `    ` T n and E  End C p T q . F or i P J 1 , n K , S i denotes the simple E -mo dule orresp onding to T i (the head of Hom C p T i , T q ). Th us, all h yp othesis to apply results of [FK℄ hold. Notation 3.43 ([FK℄) . F or L, N P mo d E , let x L, N y τ  dim k Hom B p L, N q  dim k Ext 1 B p L, N q ; x L, N y 3  3 ¸ i  0 p 1 q i dim k Ext i B p L, N q . R emark 3.44 . One k eeps the notation x , y 3 in tro dued in [FK℄, but, here, as the global dimension of E is less than 3 , this form is the Euler form (see also [ FK, remark 2.4℄). Notation 3.45. If L P mo d E , dim L denotes its image in K 0 p E q , that is its dimension v etor relativ ely to the S i . 38 Prop osition 3.46 ([FK , prop osition 2.1℄) . If L, N P mo d E , then x L, N y 3 dep ends only on dim L and N . Notation 3.47. Using previous prop osition, if L, N P mo d E , one an set x dim L, N y 3  x L, N y 3 . Dene π to b e the follo wing anonial pro jetion: π : Q  x  1 i  i P J 1 ,n K Ñ Q  x  1 i  i P I x i ÞÑ x i . Denition 3.48. F or X P C , dene the Lauren t p olynomial P X of Q  x  1 i  i P I b y P X  π p X 1 X q where X 1 X is the Lauren t p olynomial of Q  x  1 i  i P I dened b y F u and Keller in [FK ℄. In other w ords, (6) P X   n ¹ i  1 x x Hom C p T ,X q ,S i y τ i  ¸ e P N J 1 ,n K  χ p Gr e p Ext 1 C p T , X qqq n ¹ i  1 x x e,S i y 3 i  , where Gr e p Ext 1 C p T , X qq is the v ariet y of E -submo dules B of Ext 1 C p T , X q su h that dim B  e and χ is the Euler  harateristi with resp et to étale ohomology with prop er supp ort. Lemma 3.49. The L aur ent p olynomial P X dep ends only on the lass of X mo dulo Γ . Pr o of. As T is Γ -in v arian t, for ev ery g P Γ , Hom C p T , g b X q  Hom C p g b T , g b X q  g  1 b Hom C p T , X q where mo d E is anonially endo w ed with the ation of Γ indued b y the ation of Γ on C . As a onsequene, x Hom C p T , g b X q , S i y τ  x g  1 b Hom C p T , X q , S i y τ  x Hom C p T , X q , g b S i y τ whi h leads to the onlusion onerning the rst fator of the righ t-hand side of (6 ). In the same w a y , one gets Gr e p Ext 1 C p T , g b X qq  Gr g  e p Ext 1 C p T , X qq and x e, S i y 3  x g  e, g b S i y 3 whi h yields the onlusion onerning the seond fator of the righ t-hand side of ( 6 ).  A ording to lemma 3.49, it mak es sens to denote P X  P X where X is the lass of X mo dulo Γ . Here is the analogous of theorem [FK, theorem 2.2℄: Theorem 3.50. (i) F or i P I , P T i  x i . (ii) If X, Y P C , P X ` Y  P X P Y . (iii) If X, Y P C and dim Ext 1 C p X, Y q  1 , and if one xes two non split admissible short exat se quen es 0 Ñ X Ñ Z Ñ Y Ñ 0 and 0 Ñ Y Ñ Z 1 Ñ X Ñ 0 then P X P Y  P Z  P Z 1 . Pr o of. This follo ws from [FK, theorem 2.2℄ b y applying the ring morphism π .  Corollary 3.51. The P X satisfy the mutation formulas of F omin and Zelevinsky en o de d by the exhange matri es B of denition 3.36. In other wor ds, if T 1 0 P Add p C Γ q mod k r Γ s and X, Y P C ar e inde  omp osable obje ts suh that  X, Y R T 1 0 ,  add p T 1 0 , X r Γ sq is maximal mo d k r Γ s -stable rigid, 39  µ X p T 1 q  add p T 1 0 , Y r Γ sq wher e T 1  add p T 1 0 , X r Γ sq , then (7) P X P Y  ¹ i P I | B p T 1 q T 1 i X  0 P  B p T 1 q T 1 i X T 1 i  ¹ i P I | B p T 1 q T 1 i X ¡ 0 P B p T 1 q T 1 i X T 1 i . Pr o of. There exists X 0 P X r Γ s and Y 0 P Y r Γ s whi h are neigh b ours. Hene, one an supp ose that Ext 1 C p X, Y q  1 up to replaing Y b y a dieren t represen tativ e of its equiv alene lass mo dulo Γ . Let 0 Ñ X f Ý Ñ Z g Ý Ñ Y Ñ 0 b e a non split admissible short exat sequene. In this ase, f is a minimal left T 1 0 -appro ximation and g is a minimal righ t T 1 0 -appro ximation using prop osition 3.14 . Hene, for i P J 1 , n K , the n um b er of T 1 i app earing in Z is # t q P Q 1 1 | s p q q  X, t p q q  T 1 i u where Q 1 is the Auslander-Reiten quiv er of T 1 . Th us, for i P I , the n um b er of T 1 i with i P i whi h app ear in Z is # t q P Q 1 1 | s p q q  X, t p q q P T 1 i u and, as T 1 has no Γ -lo ops, if this n um b er is stritly p ositiv e, it is equal b y denition to  B p T 1 q T 1 i X , whi h p ermits to onlude. The seond term of the righ t-hand side of (7 ) an b e handled in the same w a y .  Denote b y A p C , Γ , T q the subalgebra of Q p x i q i P I generated b y the P X where X go es o v er all Γ -orbits of ob jets of C su h that À X P X X is rigid. Denote b y A p C , T q the subalgebra of Q p x i q i P J 1 ,n K generated b y the X 1 X where X go es o v er the rigid ob jets of C . Denote b y A 0 p C , Γ , T q the subalgebra of Q p x i q i P I generated b y the P X where X go es o v er the Γ -orbits of ob jets of C . Denote b y A 0 p C , T q the subalgebra of Q p x i q i P J 1 ,n K generated b y the P 1 X where X go es o v er C . Corollary 3.52. Ther e is a  ommutative diagr am of inlusions A p B p T qq   / /  _   π  A p r B p T qq   _   A p C , Γ , T q   / /  _   π p A p C , T qq  _   A 0 p C , Γ , T q π p A 0 p C , T qq . Pr o of. First of all, the inlusions A p B p T qq  A p C , Γ , T q and A p r B p T qq  A p C , T q ome from orollary 3.51. The b ottom equalit y is lear using the denition of P X in terms of X 1 X . The horizon tal middle inlusion omes from the fat that for all P X where À X P X X is rigid, X P X is rigid and therefore P 1 X P A p C , T q . The upp er horizon tal inlusion omes from the fat that if P X and P Y are link ed b y a string of m utations in A p C , Γ , T q , then P 1 X and P 1 Y are also in A p C , T q aording to prop osition 3.40 .  R emark 3.53 . In general, it seems to b e a diult problem to understand whi h of the inlusions in the previous diagram are isomorphisms. Let ∆ (resp. ∆  , r ∆ ) b e the non-orien ted v ersion of the graph whose adjaeny matrix is the upp er square submatrix of B p T q (resp. B  p T q , r B p T q ). A ording to prop osition 3.40 , r ∆ and ∆ are related b y a lassial folding pro ess. On the other hand, ∆  and ∆ are related b y a folding pro ess deformed b y some p ositiv e in teger o eien ts (the n X  ℓ p X q{ # X ). Lemma 3.54. The fol lowing ar e e quivalent: 40 (i) ∆ is a Dynkin diagr am; (ii) ∆  is a Dynkin diagr am. Mor e over, under these assumptions, r ∆ is also a Dynkin diagr am. Pr o of. Ev ery diagram an b e supp osed to b e onneted without loss of generalit y . The pro of that (ii) implies (i) and that r ∆ is a Dynkin diagram an b e done b y a nite n um b er of omputations (see table of page 45 ). Let us no w sho w that (i) implies (ii). Supp ose that ∆ is a Dynkin diagram. Let us all ritial p oin t of ∆ or r ∆ ev ery v ertex of v aluation at least 3 or ev ery non simple edge. As ∆ is a Dynkin diagram, it has at most one ritial p oin t. If r ∆ has a yle, then it indues in ∆ a yle or at least to ritial p oin ts, hene r ∆ is a tree. Moreo v er, an orbit under Γ of ritial p oin ts in r ∆ yields a ritial p oin t in ∆ . Th us, there is at most su h an orbit. Supp ose that there is t w o distint p oin ts A and B in this orbit. As r ∆ is a tree, there is a unique shortest path b et w een these t w o p oin ts, whi h is folded b y the elemen t of Γ whi h sends A on B . Hene it is easy to see that the middle of this path giv es rise to a lo op or a seond ritial p oin t in ∆ , whi h is not p ossible. Finally , r ∆ has at most one ritial p oin t, whi h leads to the onlusion using a ase b y ase pro of.  Prop osition 3.55. The fol lowing ar e e quivalent: (i) A p B p T qq has a nite numb er of luster variables; (ii) A p r B p T qq has a nite numb er of luster variables; (iii) A p B  p T qq has a nite numb er of luster variables; (iv) C has a nite numb er of isomorphism lasses of rigid inde  omp osable obje ts. Pr o of. First of all, it is lear that ( ii) or (iii ) imply (i) using prop osition 3.40. No w, if (i) is true, using the results of [FZ2 ℄, B p T q is m utation-equiv alen t to a matrix, whose prinipal square submatrix eno des a Dynkin diagram. Up to  hanging T , one an supp ose that the prinipal square submatrix of B p T q eno des a Dynkin diagram. By lemma 3.54 , the prinipal square submatries of r B p T q and B  p T q eno de Dynkin diagrams, whi h leads to (ii) and (iii ) using again [FZ2℄. It is lear that (iv) implies the three others. If (ii ) is satised, the prinipal square submatrix of r B p T qq an b e supp osed to eno de a Dynkin diagram. As a onsequene, using a theorem of Keller and Reiten [KR℄, the stable ategory of C is equiv alen t to a luster ategory , whi h yields (iv).  3.6. Linear indep endene of luster monomials. One retains the notation of setion 3.5 . Again, this setion generalizes results of [FK℄. Denition 3.56. T w o ob jets X , Y of C are said to b e  ongruent mo dulo Γ if there exists t w o deomp ositions in to indeomp osable diret summands X  X 1 ` X 2 `    ` X m and Y  Y 1 ` Y 2 `    ` Y m su h that for all i P J 1 , m K , X i and Y i are equiv alen t mo dulo Γ (that is, if there exists g P Γ su h that g b X i  Y i ). Equiv alen tly , X and Y are ongruen t mo dulo Γ if à g P Γ g b X  à g P Γ g b Y . If X P C , there exists an admissible short exat sequene 0 Ñ T 1 X Ñ T 2 X f Ý Ñ X Ñ 0 where f is a minimal righ t T -appro ximation. Using lemma 3.6 and the maximalit y of T , one gets that T 1 X P T . F ollo wing [P al ℄ and [FK℄, write ind T p X q  r T 0 X s  r T 1 X s P K 0 p T q . Denote also b y ind 1 T p X q the image of ind T p X q in K 0 p T q{ Γ . The follo wing lemma generalizes [DK , lemma 2.1℄: 41 Lemma 3.57. If X is rigid and 0 Ñ T 1 X h Ý Ñ T 2 X f Ý Ñ X Ñ 0 is the pr evious admissible short exat se quen e, then the orbit of h under the ation of Aut C p T 1 X q  Aut C p T 2 X q is a dense op en subset of Hom C p T 1 X , T 2 X q . Pr o of. Let h 1 : T 1 X Ñ T 2 X b e a morphism. Applying Hom C p , X q to the admissible short exat sequene leads to the follo wing long exat sequene: 0 Ñ Hom C p X, X q Ñ Hom C p T 2 X , X q Ñ Hom C p T 1 X , X q Ñ Ext 1 C p X, X q  0 whi h sho ws that there exists α : T 2 X Ñ X su h that αh  f h 1 . Moreo v er, as f is a righ t T -appro ximation, there exists β : T 2 X Ñ T 2 X su h that α  f β . Hene f p β h  h 1 q  0 and therefore, as h is a k ernel of f , there exists Γ : T 1 X Ñ T 1 X su h that β h  h 1  h Γ . In other w ords, h 1 is in the image of End C p T 1 X q  End C p T 2 X q Ñ Hom C p T 1 X , T 2 X q p Γ , β q ÞÑ β h  h Γ whi h is the dieren tial of the appliation Aut C p T 1 X q  Aut C p T 2 X q Ñ Hom C p T 1 X , T 2 X q p g 1 , g 2 q ÞÑ g 2 hg  1 1 . Here, w e use the t w o iden tiations Lie p Aut C p T 1 X q  Aut C p T 2 X qq  End C p T 1 X q  End C p T 2 X q and Lie p Hom C p T 1 X , T 2 X qq  Hom C p T 1 X , T 2 X q . Hene, one gets the onlusion.  The follo wing lemma is inspired from [DK , lemma 2.2℄: Lemma 3.58. If X is rigid and 0 Ñ T 1 X h Ý Ñ T 2 X f Ý Ñ X Ñ 0 is the admissible short exat se quen e dene d as b efor e, then T 1 X and T 2 X have no  ommon dir e t summand. Pr o of. Supp ose that T 1 X  T 0 ` r T 1 X and T 2 X  T 0 ` r T 2 X with T 0  0 . Using previous lemma, the orbit of h under the ation of Aut C p T 1 X q  Aut C p T 2 X q is a dense op en subset of Hom C p T 1 X , T 2 X q . As a onsequene, up to the ation of Aut C p T 1 X q  Aut C p T 2 X q , h an b e supp osed to b e of the form  h 11 h 12 h 21 h 22  where h 11 is an automorphism of T 0 . Then, using the ation of  Id T 0  h  1 11 h 12 0 Id T 1 X  ,  Id T 0 0  h 21 h  1 11 Id T 2 X   P Aut C p T 1 X q  Aut C p T 2 X q h an b e deomp osed as a diret sum and therefore f is not minimal.  The follo wing lemma is inspired from [FK , lemma 3.2℄ and [DK, theorem 2.3℄: Lemma 3.59. If X P C is rigid, then the  ongruen e lass of X mo dulo Γ is determine d by ind 1 T p X q . In other wor ds, if Y P C is rigid and if ind 1 T p Y q  ind 1 T p X q , then X and Y ar e  ongruent mo dulo Γ . 42 Pr o of. Let 0 Ñ T 1 X Ñ T 2 X f Ý Ñ X Ñ 0 and 0 Ñ T 1 Y Ñ T 2 Y f 1 Ý Ñ Y Ñ 0 b e the admissible short exat sequenes whi h dene ind T p X q and ind T p Y q . Using lemma 3.58 , T 1 X and T 2 X on one hand and T 1 Y and T 2 Y on the other hand ha v e no ommon indeomp osable summand. As a onsequene, ind T p X q and ind T p Y q fully determine them and ind 1 T p X q  ind 1 T p Y q determine their ongruene lasses mo dulo Γ . Summing up, à g P Γ g b T 1 X  à g P Γ g b T 1 Y and à g P Γ g b T 2 X  à g P Γ g b T 2 Y . If one denotes T 1  T 1 X and T 2  T 2 X , one gets the t w o follo wing admissible short exat sequenes: 0 Ñ à g P Γ g b T 1 h Ý Ñ à g P Γ g b T 2 Ñ à g P Γ g b X Ñ 0 and 0 Ñ à g P Γ g b T 1 h 1 Ý Ñ à g P Γ g b T 2 Ñ à g P Γ g b Y Ñ 0 . As À g P Γ g b X and À g P Γ g b Y are rigid, using lemma 3.57 , h and h 1 are in the same orbit under the ation of Aut C p T 1 q  Aut C p T 2 q and, as a onsequene, à g P Γ g b X  à g P Γ g b Y whi h implies that X and Y are ongruen t mo dulo Γ .  Let us no w adapt [FK, orollary 4.4℄: Prop osition 3.60. If the r ank of r B p T q is ful l, then (i) X ÞÑ P X indu es an inje tion fr om the set of inde  omp osable lasses of C mo dulo Γ suh that À X P X X is rigid. (ii) Supp ose that E  Add p C q Γ is a nite multiset of maximal rigid Γ -stable  ate gories, and x for every T 1 P E , an obje t X T 1 P T 1 . If the X T 1 ar e not  ongruent mo dulo Γ p airwise, then the P X T 1 ar e line arly indep endent. Pr o of. The rst p oin t is a diret onsequene of the seond. Supp ose that for some c T 1 P Q , ¸ T 1 P E c T 1 P X T 1  0 . Using the pro of of [FK, orollary 4.4℄, one dedues the relation ¸ T 1 P E c T 1 n ¹ i  1 x r ind T p X T 1 q : r T i ss i  0 where r ind T p X T 1 q : r T i ss is the o eien t of r T i s in the deomp osition of ind T p X T 1 q in the basis tr T j su j P J 1 ,n K . Hene ¸ T 1 P E c T 1 ¹ i P I x r ind 1 T p X T 1 q : r T i ss i  0 where r T i s is the orbit of r T i s in K 0 p T q{ Γ . Using previous lemma, the ind 1 T p X T 1 q are distint. Hene, ev ery c T 1 v anishes.  Corollary 3.61. If r B p T q is of ful l r ank then the P X , wher e X runs over the e quivalen e lasses of C mo dulo Γ suh that à X P X X is rigid, ar e line arly indep endent over Q . The luster monomials with  o eients ar e line arly indep endent over Q . Equivalently, the luster monomials without  o eients ar e line arly inde- p endent over the gr ound ring. 43 Pr o of. The seond part is an immediate onsequene of the rst one b eause the luster mono- mials ome from su h X through the inlusion A p B p T qq  A p C , Γ , T q . The rst p oin t is a lear onsequene of prop osition 3.60 .  R emark 3.62 . Of ourse, in prop osition 3.60 and orollary 3.61 , one an replae the assumption that r B p T q has maximal rank b y the stronger h yp othesis that B p T q has maximal rank. 4. Applia tions 4.1. Reminder ab out ro ot systems and en v eloping algebras. F or more details ab out this setion, see for example [Hub1, 1.4℄. Let r C b e a symmetrizable generalized Cartan matrix with ro ws indexed b y r ∆ 0 and let r ∆ b e the bi-v alued unorien ted graph with v ertex-set r ∆ 0 su h that if i, j P r ∆ 0 , there is an edge b et w een i and j if C ij  0 and its v aluation is p C ij ,  C j i q . Let Γ b e a group ating on r ∆ in su h a w a y that r ∆ has no edge b et w een an y t w o v erties of the same Γ -orbit (the ation will b e said to b e admissible). Denition 4.1. One will denote b y ∆ the unorien ted graph with v ertex-set r ∆ 0 { Γ and Cartan matrix dened b y C ij  1 # j ¸ p i,j qP i  j r C ij . Lemma 4.2. Every symmetrizable Cartan matrix  an b e obtaine d by this metho d fr om a sym- metri Cartan matrix and a yli gr oup. Pr o of. Supp ose that C 1 is a symmetrizable matrix of order n . Let p d i q i P J 1 ,n K b e the p ositiv e in teger en tries of a diagonal matrix D su h that D C 1 is symmetri. F or i P J 1 , n K put n i  ± j  i d j . Let I  ¤ i P J 1 ,n K t i u  Z { n i Z . One denotes b y a the automorphism of I dened b y p i, j q ÞÑ p i, j  1 q . F or p i, j q and p i 1 , j 1 q in I dene C 2 p i,j q , p i 1 ,j 1 q  # d i C 1 ii 1 d i _ d i 1  d i 1 C 1 i 1 i d i _ d i 1 if n i ^ n i 1 | j  j 1 0 else. It is easy to  he k that the group generated b y a ats on the diagram asso iated to the Cartan matrix C 2 . Moreo v er, it is an easy omputation to  he k that the symmetri Cartan matrix obtained from C 2 and a is C 1 .  R emark 4.3 . An other pro of of lemma 4.2 is giv en in [Lus2 , prop osition 14.1.2℄. Let g and r g b e the Ka-Mo o dy Lie algebras asso iated to C and r C . One denotes b y p r e i q i P Q 0 and p r f i q i P Q 0 (resp. p e i q i P Q 0 { Γ and p f i q i P Q 0 { Γ ) the Chev alley generators of r g (resp. g ). One sets h i  r e i , f i s and r h i  r r e i , r f i s . Let n (resp. n  ) b e the nilp oten t subalgebra (resp. opp osite nilp oten t subalgebra) of g generated b y the p e i q i P Q 0 { Γ (resp. b y the p f i q i P Q 0 { Γ ). Let b (resp. b  ) b e the Borel subalgebra (resp. opp osite Borel subalgebra) of g generated b y the p h i q i P Q 0 { Γ and n (resp. n  ). Let r n (resp. r n  ) b e the nilp oten t subalgebra (resp. opp osite nilp oten t subalgebra) of r g generated b y the p r e i q i P Q 0 (resp. b y the p r f i q i P Q 0 . Let r b (resp. r b  ) b e the Borel subalgebra (resp. opp osite Borel subalgebra) of r g generated b y the p r h i q i P Q 0 and r n (resp. r n  ). As Γ ats on r ∆ , it ats also on r g b y extending its ation on the Chev alley generators. Prop osition 4.4 ([Hub1, theorem 7.1.5℄) . Ther e is a monomorphism of Lie algebr as g ã Ñ r g Γ e i ÞÑ ¸ i P i r e i f i ÞÑ ¸ i P i r f i h i ÞÑ ¸ i P i r h i whih  an b e r estrite d to a monomorphism n ã Ñ r n Γ . If C , or e quivalently r C , is of Dynkin typ e, this monomorphism is an isomorphism. 44 Corollary 4.5. Ther e is an epimorphism κ : U p r n q  gr { Γ ։ U p n q  gr wher e the quotient by Γ has to b e understo o d as the quotient by the ide al gener ate d by the elements of the form p g f  f q for g P Γ and f P U p r n q  gr . Her e, U p r n q  gr and U p n q  gr denote gr ade d dual sp a es. If C is of Dynkin typ e, κ is an isomorphim. Pr o of. It is a lear translation of prop osition 4.4.  Lemma 4.6 ([Hub2 , prop osition 4℄) . L et R b e a r o ot system of typ e r ∆ . L et V b e the latti e gener ate d by R . Then the line ar map α : V Ñ V v ÞÑ ¸ g P Γ g v maps R to a r o ot system of typ e ∆ . 4.2. Sub I J and partial ag v arieties. This appliation generalizes [GLS6 ℄. Let Q b e a quiv er su h that the underlying unorien ted graph r ∆ is a Dynkin diagram of t yp e A , D or E . Let Γ b e a group ating on Q in su h a w a y that Q has no arro w b et w een an y t w o v erties of the same orbit (the ation will b e said to b e admissible). It indues an ation on r ∆ . W e denote b y Q Γ the same quiv er as in setion 2.8 and b y r ∆ Γ the underlying unorien ted graph. W e denote b y ∆ the diagram dened in setion 4.1. Here is the list of all p ossible ases, where Γ ats faithfully on a Dynkin diagram r ∆ . r ∆ Γ ∆ r ∆ Γ 1 2 . . . n  1 L L L L n 1 1 2 1 . . . p n  1 q 1 t t t t Z { 2 Z 1 2 . . . n  1 < n n  1 2 . . . n  1 t t t J J J n  n 1 2 . . . n  1 u u u u H H H n 1 Z { 2 Z 1 2 . . . n  1 > n 1  2  . . . p n  1 q  L L L L n 1  2  . . . p n  1 q  r r r r 1 @ @ @ 1 1 2 1 2    Z { 3 Z 1 < 2 2 1 1 w w w w w D D D D 2 e 2 iπ 3 2 e  2 iπ 3 1 @ @ @ 1 1 2 1 2    S 3 1 < 2 2  1  { { { D D D 2 2 1  z z z C C C 2  1 B B B 2 @ @ @ 3 4 2 1    1 1 ~ ~ ~ Z { 2 Z 1 2 < 3 4 4  3  { { { 1 2 } } } A A A 3  C C C 4  45 R emark 4.7 . Observ e that all non simply-laed Dynkin diagrams an b e realized for appropriate r ∆ and Γ . One retains the notation of setion 4.1. Let N and r N b e the Lie groups asso iated with n and r n . Notation 4.8. If i P Q 0 , x i denotes the one-parameter subgroup of r N dened b y x i p t q  exp p te i q . Let X P mo d Λ Q and p i q  i 1 i 2 . . . i n b e a w ord on Q 0 . One will denote b y Φ X, p i q the (losed) sub v ariet y of Gr 0 p X q  Gr 1 p X q      Gr n  2 p X q  Gr n  1 p X q onsisting of the p X 0 , X 1 , . . . , X n  2 , X n  1 q su h that  for all j P J 1 , n  1 K , X j  1  X j ;  for all j P J 1 , n  1 K , X j { X j  1  S i j ;  X { X n  1  S i n . The follo wing result is obtained b y dualit y from the Lagrangian onstrution of U p n q b y Lusztig [Lus1 ℄, [Lus4 ℄. Theorem 4.9 ([GLS3 , 9℄) . F or al l X P mo d Λ Q , ther e exists a unique ϕ X P C r N s suh that for every wor d p i q on Q 0 , and every t P C n (wher e n is the length of p i q ), ϕ X p x i 1 p t 1 q x i 2 p t 2 q . . . x i n p t n qq  ¸ a P N n χ  Φ X, p i q a  t a a ! . Her e, p i q a  i 1 . . . i 1 lo omo on a 1 i 2 . . . i 2 lo omo on a 2 . . . i n . . . i n lo omo on a n , t a  t a 1 1 t a 2 2 . . . t a n n and a !  a 1 ! a 2 ! . . . a n ! . Using the dualities C r N s  U p n q  gr and C r r N s  U p r n q  gr , w e an lift the ation of Γ on U p r n q  gr to an ation of Γ on C r r N s . Therefore the isomorphism κ dened in orollary 4.5 an b e lifted to an isomorphism κ : C r r N s{ Γ  C r N s where the quotien t b y Γ is the quotien t b y the ideal generated b y the elemen ts of the form p g f  f q for g P Γ and f P C r r N s . Notation 4.10. F or X P mo d Λ Q , let ψ X  κ p π p ϕ X qq where π : C r r N s Ñ C r r N s{ Γ is the anonial pro jetion. The ation of Γ on Q indues an ation of Γ on k Q , then on Λ Q , then on mo d Λ Q (see setion 2.8 ). F ollo wing the pro of of [GLS5, theorem 3, 8℄, it is easy to see that the follo wing diagram omm utes: Ext 1 Λ Q p 1 ,  2 q c   g b / / Ext 1 Λ Q p g b  1 , g b  2 q c   Ext 1 Λ Q p 2 ,  1 q  Ext 1 Λ Q p g b  2 , g b  1 q  p g bq  o o where c is the funtorial isomorphism from Ext 1 Λ Q p 1 ,  2 q to Ext 1 Λ Q p 2 ,  1 q  . In other terms, the ation of Γ on mo d Λ Q is 2 -Calabi-Y au in the sense of denition 2.46 . Lemma 4.11. F or X P mo d Λ Q and g P Γ , ψ g b X  ψ X . Pr o of. One has ϕ g b X p x i 1 p t 1 q x i 2 p t 2 q . . . x i n p t n qq  ¸ a P N n χ  Φ g b X, p i q a  t a a !  ¸ a P N n χ  Φ X,g  1 p i q a  t a a !  ϕ X p x g  1  i 1 p t 1 q x g  1  i 2 p t 2 q . . . x g  1  i n p t n qq 46 whi h implies the result.  Notation 4.12. One will denote ψ X  ψ X where X is the Γ -orbit of X . Theorem 4.13 ([GLS2 , lemma 7.3℄ and [GLS3 , theorem 9.2℄) . (i) If X, Y P mo d Λ Q then ϕ X ` Y  ϕ X ϕ Y . (ii) If X, Y P mo d Λ Q and dim Ext 1 Λ Q p X, Y q  1 , and if one  onsiders two non-split short exat se quen es 0 Ñ X Ñ Z Ñ Y Ñ 0 and 0 Ñ Y Ñ Z 1 Ñ X Ñ 0 then ϕ X ϕ Y  ϕ Z  ϕ Z 1 . Corollary 4.14. (i) If X, Y P mo d Λ Q , ψ X ` Y  ψ X ψ Y . (ii) If X, Y P mo d Λ Q and dim Ext 1 Λ Q p X, Y q  1 , and if one  onsiders two non-split short exat se quen es 0 Ñ X Ñ Z Ñ Y Ñ 0 and 0 Ñ Y Ñ Z 1 Ñ X Ñ 0 then ψ X ψ Y  ψ Z  ψ Z 1 . Pr o of. This follo ws immediately from the fat that ψ X is the image of ϕ X under a ring homo- morphism.  Let no w J  Q 0 b e non-empt y , Γ -stable and K  Q 0 z J . Denition 4.15. F or j P Q 0 , one denotes b y I j the injetiv e Λ Q -mo dule of so le S j . Put I J  à j P J I j and denote b y Sub I J the full sub ategory of mo d Λ Q whose ob jets are isomorphi to submo dules of I ` n J for some n P N . Prop osition 4.16 ([GLS6 , 3℄) . The  ate gory Sub I J is an exat, Hom -nite, Krul l-Shmidt, F r ob enius, and 2 -Calabi-Y au sub  ate gory of mo d Λ Q . Lemma 4.17. A l l pr oje tive obje ts of Sub I J have right rigid quasi-appr oximations. Pr o of. First of all, for ev ery simple Λ Q -mo dule S , à g P Γ g b S is rigid b eause the ation of Γ on Q is admissible. Moreo v er, the injetiv e ob jets L i of Sub I J onstruted expliitly in [GLS6 , 3.3℄ ha v e simple heads. It is no w lear that lemma 3.20 an b e applied.  Notation 4.18. If i P Q 0 , one denotes b y E : i the funtor from mo d Λ Q to itself that maps a mo dule to its quotien t b y the largest p ossible p o w er of S i . Prop osition 4.19 ([GLS6 , prop osition 5.1℄) . The E : i satisfy the fol lowing r elations (i) E : i E : i  E : i ; (ii) E : i E : j  E : j E : i if ther e is no e dge b etwe en i and j in ∆ ; (iii) E : i E : j E : i  E : j E : i E : j if ther e is an e dge b etwe en i and j in ∆ . 47 Notation 4.20. If i P Q 0 { Γ , one denotes E : i  ¹ i P i E : i whi h is w ell-dened b eause the fators E : i in the pro dut omm ute. This funtor maps the Γ -stable ob jets of mo d Λ Q to Γ -stable ob jets. Denote r ∆ K (resp. ∆ K ) the restrition of the diagram r ∆ to K (resp. of the diagram ∆ to K { Γ ). Denote b y W ,  W , W K and  W K the W eyl groups of ∆ , r ∆ , ∆ K and r ∆ K . One denotes b y p σ i q i P Q 0 (resp. p σ i q i P Q 0 { Γ ) the generators of  W (resp. W ). One gets an injetiv e morphism W Ñ  W σ i ÞÑ ¹ i P i σ i whi h restrits to a morphism from W K to  W K . Prop osition 4.21. L et p i q b e a r e du e d expr ession of the longest element of W and ℓ b e its length. Assume that p i q has a left fator whih is a r e du e d expr ession of the longest element of W K . Then T p i q  ℓ à m  1 E : i 1 E : i 2 . . . E : i m  à i P i m I i  ` à i P Q 0 I i has a dir e t summand T p i q ,K whih is maximal rigid and Γ -stable in Sub I J . Pr o of. The only thing to add to [GLS6 , prop osition 7.3℄ is that T p i q ,K is Γ -stable. It is lear b y denition of the funtors E : i .  Example 4.22. Supp ose here that r ∆  A 5 is indexed in the follo wing w a y: a b c b 1 a 1 on whi h Γ  Z { 2 Z ats in the only non-trivial p ossible w a y . Hene ∆  C 3 indexed in the follo wing w a y: a b < c Let J { Γ  t c u . Then p i q  p a, b, a, c, b, a, c, b, c q is suitable. Then, it is easy to ompute T p i q and therefore T p i q ,K (see gure 1 ). Fix no w T  T p i q ,K as in prop osition 4.21. Let T  add p T q . Lemma 4.23. The  ate gory T has no Γ -lo ops nor Γ - 2 -yles. Pr o of. By [GLS6 ℄, T is luster-tilting and End Sub I J p T q is of nite global dimension. The result follo ws b y theorem 3.33 .  Hene, w e an apply the results of setion 3.5 . W e retain the notation of setion 3.5. Example 4.24. Con tin ue with example 4.22 . The Auslander-Reiten quiv er of T is displa y ed in gure 2. The ation of Γ orresp onds to the reetion in the middle horizon tal τ -orbit. The pro jetiv e-injetiv e ob jets are the v e righ tmost ones. Indexing the lines of the ex hange matrix b y the Γ -orbits of c       @ @ b   = = b 1   ~ ~ c , b 1   ~ ~ c , c , a 1   ~ ~ b 1   ~ ~ c , b 1       @ @ c        = = a 1   ~ ~ b   = = b 1   ~ ~ c , c       @ @ b        ; ; ; b 1       @ @ a   < < < c        = = a 1   ~ ~ b   = = b 1   ~ ~ c 48 T p i q  a   ? ? b   > > c   @ @ b 1 ` a 1   ~ ~ b 1   ~ ~ c     b ` b       > > a   > > c       @ @ b   = = b 1   ~ ~ c ` b 1       @ @ c        = = a 1   ~ ~ b   = = b 1   ~ ~ c ` a   ? ? b   > > c ` a 1   ~ ~ b 1   ~ ~ c ` c       @ @ b   = = b 1   ~ ~ c ` b   > > c ` b 1   ~ ~ c ` c ` a   ? ? b   > > c   @ @ b 1   ? ? c 1 ` b       > > a   > > c       @ @ b   ; ; ; b 1       @ @ c   = = a 1   ~ ~ b 1 ` c       @ @ b        ; ; ; b 1       @ @ a   < < < c        = = a 1   ~ ~ b   = = b 1   ~ ~ c ` b 1       @ @ c        = = a 1   ~ ~ b       = = b 1   ~ ~ a   ? ? c     b ` a 1   ~ ~ b 1   ~ ~ c     b     a T p i q ,K  b       > > a   > > c       @ @ b   = = b 1   ~ ~ c ` b 1       @ @ c        = = a 1   ~ ~ b   = = b 1   ~ ~ c ` a   ? ? b   > > c ` a 1   ~ ~ b 1   ~ ~ c ` c       @ @ b   = = b 1   ~ ~ c ` b   > > c ` b 1   ~ ~ c ` c ` c       @ @ b        ; ; ; b 1       @ @ a   < < < c        = = a 1   ~ ~ b   = = b 1   ~ ~ c . Figure 1. Expliit omputation of T p i q and T p i q ,K in this order, one gets B p T q          0  1 1 2 0  2  1 1 0 0  1 0  2 1 0 1 0 0  Æ Æ Æ Æ Æ Æ  . F or X P mo d Λ Q and i P Q 0 { Γ , let k i p X q b e the total dimension of the maximal submo dule of X supp orted b y i . Let R b e the set of isomorphism lasses of Λ Q -mo dules X su h that à g P Γ g b X is rigid. If d P N Q 0 { Γ , R d is the set of elemen ts of R of dimension v etor (summed on Γ -orbits) d and if k P N , R d , i ,k  t X P R d | k i p X q  k u . One will denote b y  the equiv alene relation 49 a   ? ? b   > > c b   > > c b       > > a   > > c       @ @ b   = = b 1   ~ ~ c c c       @ @ b   = = b 1   ~ ~ c c       @ @ b        ; ; ; b 1       @ @ a   < < < c        = = a 1   ~ ~ b   = = b 1   ~ ~ c b 1   ~ ~ c b 1       @ @ c        = = a 1   ~ ~ b   = = b 1   ~ ~ c a 1   ~ ~ b 1   ~ ~ c        ✒        ✒        ✒      ✒      ✒      ✒ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❅ ❅ ❘ ✛ ✛ ✛ ✛ Figure 2. Auslander-Reiten quiv er of T on R iden tifying X and Y if à g P Γ g b X  à g P Γ g b Y . Lemma 4.25. With the pr evious notation, E : i indu es an inje tion fr om R d , i ,k {  to R d  k i , i , 0 {  . Pr o of. First of all, the map is learly dened. Let no w X, Y P R d , i ,k su h that E : i p X q  E : i p Y q . Dene r X  À g P Γ g b X and r Y  À g P Γ g b Y . These t w o rigid mo dules ha v e the same dimension v etor. Moreo v er, at ea h v ertex of i , they ha v e the same so le. Finally E : i p r X q  E : i p r Y q . Using [Lus1 , lemma 12.5 (e)℄ together with the result stating that the orbit of a rigid mo dule is a dense op en subset of its irreduible omp onen t in the mo dule v ariet y [ GLS3 , orollary 3.15℄, it implies that r X  r Y . Hene X  Y whi h is the laimed result.  F or X P mo d Λ Q , denotes b y ε X the elemen t of U p n q  orresp onding to ψ X . If f P U p n q , i P Q 0 { Γ and n P N , one gets ε X p f e n i q  ¸ k P Z k χ ! Y P Gr n p X q | dim Y  i n and ε  X { Y p f q  k )  50 where dim Y denotes the dimension v etor (summed on Γ -orbits) of Y and χ denotes the Euler  harateristi (it follo ws for example from [GLS2, 5.2℄). Prop osition 4.26. (i) F or al l X P R , ther e exists f X P U p n q homo gene ous of de gr e e d suh that for every Y P R , ε Y p f X q  " 1 if X  Y ; 0 else. (ii) The ψ X , wher e X runs over the e quivalen e lasses of inde  omp osable obje ts of T , ar e algebr ai al ly indep endent. (iii) Cluster monomials of C r N s ar e line arly indep endent. R emark 4.27 . The follo wing pro of is an adaptation of the pro of of existene of the semianonial basis b y Lusztig [Lus4℄. W e suggested a pro of in [Dem2℄ whi h w as v ery lose to this one, with a dual desription of the luster  harater. Unfortunately , w e are not able to onstrut in this w a y an analogue of the dual semianonial basis in the non simply-laed ase, but only the set of luster monomials whi h should b e a part of it. Pr o of. First, (iii ) is an easy onsequene of ( i) b eause luster monomials of C r N s are of the form ψ X where X P R . Moreo v er (ii ) is a partiular ase of (iii). Let d b e su h that X P R d . Let us onstrut f X b y indution on d . F or d  0 , f 0  1 . Supp ose that d  0 . As X is nilp oten t, there exists i su h that k i p X q ¡ 0 . Let us no w argue b y dereasing indution on k i p X q . Supp ose the result is pro v ed for ev ery X 1 of dimension d su h that k i p X q  k i p X 1 q 6 d i . Let f   f E : i p X q e k i p X q i . F or Y P R , one gets ε Y p f  q  ¸ k P Z k χ  ! Z P Gr k i p X q p Y q | dim Z  i k i p X q and ε  Y { Z  f E : i p X q   k )   $ ' & ' % 0 if k i p Y q  k i p X q ε E : i p Y q  f E : i p X q  if k i p Y q  k i p X q indeterminate if k i p Y q ¡ k i p X q  $ ' ' & ' ' % 0 if k i p Y q  k i p X q 0 if k i p Y q  k i p X q and E : i p Y q ≁ E : i p X q 1 if k i p Y q  k i p X q and E : i p Y q  E : i p X q indeterminate if k i p Y q ¡ k i p X q  $ ' ' & ' ' % 0 if k i p Y q  k i p X q 0 if k i p Y q  k i p X q and Y ≁ X 1 if Y  X indeterminate if k i p Y q ¡ k i p X q . Hene f X  f   ¸ Y P R d { Γ k i p Y q¡ k i p X q ε Y p f  q f Y is suitable (the f Y in the sum exist b y indution).  The seond p oin t of prop osition 4.26 together with orollary 4.14 pro v es that pp ψ X q , B p T qq is the initial seed of a luster algebra where X runs o v er the equiv alene lasses of indeomp osable ob jets of T . Using [GLS6 , 9.3.2℄, one an no w expliitly ompute B p T q . One denotes b y r the n um b er of p ositiv e ro ots of ∆ and r K the n um b er of p ositiv e ro ots of ∆ | K { Γ . Consider p i q and ℓ dened as b efore. Let I  Q 0 { Γ ² J 1 , ℓ K ordered in the follo wing w a y: the order on Q 0 { Γ do es not matter, the order on J 1 , ℓ K is the natural one and, if j P Q 0 { Γ and n P J 1 , ℓ K then j  n . W e no w extend the w ord p i q dened on J 1 , ℓ K to a w ord dened on I b y seting, for j P Q 0 { Γ , i j  j . Let e pp i qq  t n P J 1 , ℓ K | D m P J 1 , ℓ K , m ¡ n and i m  i n u . If 51 n P e pp i qq ² Q 0 { Γ , then one denotes n   min t m P J 1 , ℓ K | m ¡ n and i m  i n u . F or j P K { Γ , let t j  max t t 6 r K | i t  j u . F or j P J { Γ , let t j  j . Then, one onstruts a matrix B pp i q , K q whose lines are indexed b y p K r K , r K X e pp i qqq ² t t j | j P Q 0 { Γ u and olumns b y K r K , r K X e pp i qq : B pp i q , K q mn  $ ' ' ' ' & ' ' ' ' % 1 if m   n  1 if n   m  C i m i n if n  m  n   m  C i m i n if m  n  m   n  0 else. where C is the Cartan matrix of ∆ . Example 4.28. Con tin ue with example 4.22 . Then e pp i qq  t 1 , 2 , 3 , 4 , 5 , 7 u . One omputes a   1 , b   2 , c   4 , 1   3 , 2   5 , 3   6 , 4   7 , 5   8 and 7   9 . And also t a  3 , t b  2 and t c  c . One dedues that B pp i q , K q has lines indexed b y t 4 , 5 , 7 , 3 , 2 , c u and olumns indexed b y t 4 , 5 , 7 u . The Cartan matrix of ∆ is C    2  1 0  1 2  2 0  1 2   . Hene, B pp i q , K q          0  1 1 2 0  2  1 1 0 0  1 0  2 1 0 1 0 0  Æ Æ Æ Æ Æ Æ  . Prop osition 4.29. Ther e is an indexation of the Γ -orbits of inde  omp osable dir e t summands of T  T p i q ,K by t t n | n P Q 0 { Γ u ² p K r K , r K X e pp i qqq suh that, via this identi ation, B p T q  B pp i q , K q . Pr o of. Let p i q b e the image of p i q b y the substitution i P Q 0 { Γ ÞÑ ¹ i P i i where the order of the letters in the pro dut do es not matter. Hene, one gets a represen tativ e of the longest elemen t of  W . First of all, aording to [GLS6 , 9.3.2℄, r B p T q  r B pp i q , K q where r B pp i q , K q is the analogous of B pp i q , K q for the ation of the trivial group. T o n P J 1 , r K , one assigns the set n 1  J 1 , r r K where r r is the n um b er of p ositiv e ro ots of r ∆ in su h a w a y that the set of letters at p ositions n 1 in p i q omes from the letter at p osition n in p i q through the ab o v e substitution. F or j P Q 0 { Γ , let j 1  j . Finally , for n P Q 0 { Γ ² J 1 , r K , let n  P n 1 . Then, it is enough to do the follo wing 52 omputation using prop osition 3.40 , where r C is the Cartan matrix of r ∆ : B p T q mn  ¸ r m P m 1 r B p T q r mn   ¸ r m P m 1 r B pp i q , K q r mn   ¸ r m P m 1 $ ' ' ' ' & ' ' ' ' % 1 if r m   n   1 if n   r m  r C i  m i n  if n   r m  n   r m  r C i  m i n  if r m  n   r m   n  0 else.  $ ' ' ' ' & ' ' ' ' % 1 if m   n  1 if n   m ° r m P m 1  r C i  m i n  if n  m  n   m  ° r m P m 1 r C i  m i n  if m  n  m   n  0 else.  $ ' ' ' ' & ' ' ' ' % 1 if m   n  1 if n   m  C i m i n if n  m  n   m  C i m i n if m  n  m   n  0 else. whi h is the exp eted result.  Corollary 4.30. The matrix B p T q is of ful l r ank. Pr o of. It is lear that for an y olumn index n of the matrix B pp i q , K q , there is a unique line index n  su h that p n  q   n . Hene, for all olumn indies n , B pp i q , K q n  n  1 b y denition of B pp i q , K q . Moreo v er, if m  n are t w o olumn indies, then B pp i q , K q m  n  0 as p m  q   m  n . F or summarize, the submatrix of B pp i q , K q whose lines are the n  in the same order as the olumns is lo w er triangular with diagonal 1 .  R emarks 4.31 .  One onjetures that for ev ery Γ -orbit X of isomorphism lasses of Sub I J , one has ψ X  P X p ψ T i q i P Q 0 { Γ . This is learly true for the rigid X whi h an b e rea hed from T b y m utations.  Using orollary 3.61, it giv es another pro of of the linear indep endene of luster mono- mials of C r N s than prop osition 4.26 . Let G (resp. r G ) b e the onneted and simply-onneted simple Lie group orresp onding to the Lie algebra g (resp. r g ). Let B (resp. r B ) b e its Borel subgroup orresp onding to the Lie algebra b (resp. r b ). No w, N and r N are onsidered to b e unip oten t subgroups of B and r B . Let r B K b e the parab oli subgroup of r G generated b y r B and the one-parameter subgroups t x i p t q for i P K and t P C . Let also B K b e the parab oli subgroup of G generated b y B and the images in G of the one-parameters subgroups t x i p t q for i P K and t P C . Let r N K b e the unip oten t radial of r B K . Let N K b e the unip oten t radial of B K . Let A 1 p Sub I J , Γ , T q b e the subalgebra of C r N s generated b y the ψ X , where X runs o v er the Γ -orbits of isomorphism lasses of Sub I J su h that à X P X X is rigid. Let A 1 0 p Sub I J , Γ , T q b e the subalgebra of C r N s generated b y the ψ X , where X runs o v er the Γ -orbits of Sub I J . The algebras A p Sub I J , Γ , T q and A 0 p Sub I J , Γ , T q w ere dened in setion 3.5 . 53 Prop osition 4.32. If the  onje tur e of r emark 4.31 holds true then ther e is a  ommutative diagr am A p B p T qq i I v v n n n n n n n n n n n n  u ( ( Q Q Q Q Q Q Q Q Q Q Q Q A p Sub I J , Γ , T q  _    / / A 1 p Sub I J , Γ , T q  _   A 0 p Sub I J , Γ , T q  / / A 1 0 p Sub I J , Γ , T q . Pr o of. It is lear b eause the p ψ T i q i P Q 0 { Γ are algebraially indep enden t.  R emark 4.33 . In prop osition 4.32 , the four inlusions exist ev en if the onjeture of remark 4.31 do es not hold. It is a hard problem to understand when these inlusions are isomorphisms. Prop osition 4.34. One has C r N K s  A 1 0 p Sub I J , Γ , T q . Pr o of. This is the immediate translation of [GLS6, prop osition 9.1℄ together with the fat that κ : C r r N s{ Γ  C r N s dened just b efore notation 4.10 restrits to an isomorphism C r r N K s{ Γ  C r N K s .  Conjeture 4.35. One has C r N K s  A p B p T qq . Lemma 4.36. The lusters of A p B p T qq have (i) r  r K luster variables; (ii) # Q 0 { Γ  o eients. Pr o of. The p oin t (ii) is pro v ed using [GLS6 , prop osition 3.2℄. The pro of of (i) starts with the partiular ase K  H . In this ase, the upp er b ound b y r is found as in [ GS℄, b y oun ting the Γ -stable omp onen ts of the mo dule v ariet y , and using the desription of the ro ots of ∆ of lemma 4.6 . Let no w T b e a basi maximal Γ -stable rigid Λ Q -mo dule. As w e ha v e seen b efore, it is luster tilting and therefore, aording to [GLS6 , prop osition 7.3℄, it has r r indeomp osable diret summands where r r is the n um b er of p ositiv e ro ots of r ∆ . One see that the Γ -orbits of these summands orresp ond to Γ -orbits of ro ots in the desription of [ GS℄. Hene T has exatly r Γ -orbits of indeomp osable summands using lemma 4.6. After that, if K  H , the pro of of [GLS6 , prop osition 7.1℄ w orks exatly in the same w a y and therefore, there is at most r  r K luster v ariables. The fat that r  r K is rea hed is the same as ab o v e for pro ving that r is rea hed.  One an no w pro v e the follo wing result, a part of whi h is pro v ed in [GLS6, 11.4℄ and the other part is onjetured in [GLS6 , 14.2℄: Prop osition 4.37. The luster algebr a A p B p T qq has a nite numb er of lusters exatly in the fol lowing  ases (the ir le d verti es ar e those of J and n is the numb er of verti es): T yp e of G T yp e of A p B p T qq d   . . .  A 0  d   . . .  A n  2 d  d   . . .  A n  1 d   . . .  d  p A 1 q n  1 d   . . .  d   A 2 n  4 d  d   . . .  d  A 2 n  3 54 T yp e of G T yp e of A p B p T qq d   . . .  d   A 2 n  4  d  d   D 4 d  d  d   D 5 d  d  d  d  D 6   d    D 4 d   d    E 6  d  d    E 6 d  d  d    E 7   d     E 6  d  d     E 8   d      E 8  d   . . .       > > > > >  p A 1 q n  2 d         ? ? ? ? ? d  A 5 d          A A A A A  A 5 d   . . .  <  p A 1 q n  1 d   . . .  >  p A 1 q n  1 d  > d  B 2  C 2   < d  B 3   > d  C 3 Pr o of. All simply laed ases are pro v ed in [GLS6 , 11.4℄. The other ases m ust ome from a simply laed ase endo w ed with a group ation b y prop osition 3.55 . Th us, one has to lo ok at the automorphisms of ea h diagram stabilizing J . This giv es immediately a list of v e non simply-laed ases: T yp e of r G Γ T yp e of G d   . . .  d  Z { 2 Z d   . . .  <  d  d  d  Z { 2 Z d  > d  55 T yp e of r G Γ T yp e of G   d    Z { 2 Z   < d   d   . . .       > > > > >  Z { 2 Z d   . . .  >  d         ? ? ? ? ? d  Z { 2 Z   > d  Compute their luster t yp e: the diagram d   . . .  <  with n v erties omes from the diagram d   . . .  d  with 2 n  1 v erties endo w ed with the only non-trivial automorphism of order 2 . Hene, its t yp e m ust b e obtained from p A 1 q 2 n  2 with the ation of Z { 2 Z ; th us, its luster t yp e is of the form p A 1 q k for some k . Using lemma 4.36, the n um b er of luster v ariables in a luster is r  r K  n 2  p n  1 q 2  2 n  1 and if one remo v es the n o eien ts, its t yp e has to b e of rank n  1 whi h implies the its luster t yp e is p A 1 q n  1 . The other ases an b e handled b y the same metho d.  4.3. Categories C M and unip oten t groups. This appliation is a generalization of [GLS1 ℄. Let Q b e no w an arbitrary quiv er without orien ted yles. Let Γ b e a group ating on Q in an admissible w a y (see previous setion). The algebra k Q is naturally iden tied with a subalgebra of Λ Q , one denotes b y π Q : mo d Λ Q Ñ mo d k Q the orresp onding restrition funtor. It is essen tially surjetiv e. Denition 4.38. A mo dule M P mo d k Q is said to b e terminal if (i) M is preinjetiv e; (ii) if X P mo d k Q is indeomp osable and Hom k Q p M , X q  0 , then X P add p M q ; (iii) add p M q on tains all injetiv e k Q -mo dules. Denition 4.39. Let M P mo d k Q b e a terminal mo dule. Dene C M  π  1 Q p add p M qq . Theorem 4.40. [GLS1 , theorem 2.1℄ L et M P mo d k Q b e a terminal mo dule. Then the  ate gory C M is an exat, Hom -nite, Krul l-Shmidt, F r ob enius, and 2 -Calabi-Y au sub  ate gory of mo d Λ Q . Let M b e a terminal, Γ -stable mo d k Q -mo dule. Reall the follo wing lemma of Geiÿ, Leler and S hrö er: Lemma 4.41 ([GLS1 , lemma 5.6℄) . The  ate gory C M is a sub  ate gory of mo d Λ Q stable by fators. In other terms, for X P C M and Y a Λ Q -submo dule of X , then X { Y P C M . Corollary 4.42. A l l pr oje tive obje ts of C M have left rigid quasi-appr oximations. 56 Pr o of. In order to pro v e this, it is enough to see that the h yp othesis of lemma 3.20 are satised. The lemmas at the b eginning of [GLS1 , 8℄ pro v e that the pro jetiv e ob jets of C M ha v e simple so les in mo d Λ Q . Moreo v er, as the ation of Γ on Q is admissible, À g P Γ g b S is rigid for all simple Λ Q -mo dules S . The other h yp othesis of lemma 3.20 are immediate onsequenes of lemma 4.41 .  The Λ Q -mo dules T M and T _ M onstruted in [GLS1 , 7℄ are luster-tilting and End C M p T M q and End C M p T _ M q are of nite global dimension. Moreo v er, they are Γ -stable as M is. The ation of Γ is 2 -Calabi-Y au in the sense of denition 2.46 for the same reasons as in the previous setion. Hene, one an apply the results of setion 3.5 in C M with T  add p T M q and T _  add p T _ M q . Let Θ b e the Gabriel quiv er of End k Q p M q , in whi h one adds an arro w x Ñ τ p x q for ev ery v ertex x su h that τ p x q orresp ond to an indeomp osable ob jet of add p M q (where τ is the Auslander-Reiten translation). A v ertex i of Θ is alled to b e fr ozen if τ p i q R Θ . Prop osition 4.43 ([GLS1 , 7.2℄) . The matri es r B p T q and r B p T _ q ar e e qual, with an appr opriate indexation, to the adja eny matrix of Θ , fr om whih the  olumns  orr esp onding to fr ozen verti es ar e r emove d. Notation 4.44. Let N Q op 0  N  Q 0 and N Q op 1  N  Q op 1 ² N  Q 1 endo w ed with maps s, t : N Q op 1 Ñ N Q op 0 dened b y s p n, q q  " p n, s p q qq if p n, q q P N  Q op 1 p n  1 , s p q qq if p n, q q P N  Q 1 t p n, q q  p n, t p q qq . Th us, one dened a quiv er N Q op . Let also r Θ b e the quiv er N Q op , on whi h one adds the arro ws p n, i q Ñ p n  1 , i q for p n, i q P N Q op 0 . As Q has no yles, an order on Q 0 an b e xed in su h a w a y that ev ery arro w of Q has a larger target than its soure and in su h a w a y that Γ ats on Q 0 b y inreasing maps. Th us, this order indues an order on Q 0 { Γ . Then, one endo ws N Q op 0  N  Q op with the lexiographi order. It is lassial that the Auslander-Reiten quiv er of add p M q is a sub quiv er of N Q op and that Θ is a sub quiv er of r Θ , the Auslander-Reiten translation τ of add p M q b eing giv en b y the arro ws p n, i q Ñ p n  1 , i q in r Θ . Corollary 4.45. The matrix B p T q has ful l r ank. Pr o of. One will use the struture of r Θ . F or ev ery non-frozen Γ -orbit X of Θ , τ p X q is a Γ -orbit of Θ . One restrits the order on v erties of r Θ to an order on v erties of Θ . Then, one gets for ev ery non-frozen Γ -orbit X of Θ , B p T q τ p X q ,X   1 and, if one onsiders another non-frozen Γ -orbit Y  X of Θ , B p T q τ p X q ,Y  0 b y onstrution of the order on r Θ . In other w ords, the submatrix of B p T q whose lines are the τ p X q is upp er triangular of diagonal  1 .  Corollary 4.46. The luster monomial of A p C M , Γ , T q ar e line arly indep endent. Pr o of. It is a diret appliation of orollary 3.61.  Theorem 4.47. F or every ayli luster algebr a without  o eient A , ther e is a quiver Q , a nite gr oup Γ ating on Q , a terminal mo dule M of mo d k Q and a luster-tilting sub  ate gory T of C M suh that the luster algebr a A p C M , Γ , T q with  o eients sp e ialize d to 1 is isomorphi to A . This holds in p artiular for luster algebr as of nite typ e. Pr o of. Using the same pro of as for lemma 4.2 for an ayli ex hange matrix B of A , there is a sk ew-symmetri matrix r B and an ation of a nite group Γ on it, su h that B is giv en b y prop osition 3.40 (ii ). It is easy to see that r B is ayli. Let Q b e the quiv er of adjaeny matrix r B . If mo d k Q has no pro jetiv e-injetiv e ob jet (i.e., Q is not of t yp e A n orien ted in one 57 diretion), then, M  k Q op ` τ p k Q op q is suitable. If Q is of t yp e A n orien ted in one diretion, Γ ats trivially . In this ase, let Q 1 b e a quiv er of t yp e A n  1 . Then Q 1 with M  k Q 1 op ` τ p k Q op q is suitable (where Q is onsidered to b e a sub quiv er of Q 1 with the same soure).  Here are t w o results whi h are immediate onsequenes of [ GLS1 ℄. Prop osition 4.48. Ther e exists in A p C M , Γ , T q a se quen e of mutations going fr om T to T _ . Pr o of. Geiÿ, Leler and S hrö er desrib ed in [GLS1 , 18℄ a sequene of m utations going from T to T _ in A p C M , T q . It is easy to see, follo wing their algorithm, that one an sort these m utations in su h a w a y that the m utations of a Γ -orbit of C are onseutiv e (it is enough to p erm ute m utations whi h omm ute). Then, b y prop osition 3.40 , it is lear that this sequene of m utations omes from a sequene of m utations in A p C M , Γ , T q .  Prop osition 4.49. The algebr a A p B p T qq is a p olynomial ring. Mor e pr e isely, A p B p T qq  C r P X s X P  M wher e  M is the set of Γ -orbits of isomorphism lasses of inde  omp osable obje ts of add p M q . Pr o of. By orollary 3.52 , A p B p T qq  A p r B p T qq{ Γ so, using [GLS1 , theorem 3.4℄, there is an inlusion A p B p T qq  C r P X s X P  M . F or the on v erse inlusion, one uses the same te hnique as in [GLS1 , 20.2℄: ev ery X P  M app ears in the sequene of m utations of the previous prop osition.  The end of this setion deals with the ase of general Ka-Mo o dy groups. It w orks in partiu- lar for semisimple Lie groups. F or more details ab out the innite dimensinal ase, in partiular ab out the onstrutions of the nite dimensional subgroups N p w q and N w , one refers to [GLS1 ℄. One retains the notation of setion 4.1. Let N (resp. r N ) b e the pro-unip oten t pro-group dened from n (resp. r n ) as in [GLS1 , 22℄ (see also [Kum , 4.4℄) in su h a w a y that C r N s  U p n q  gr and C r r N s  U p r n q  gr . One denotes b y r ∆  M the set of dimension v etors of indeomp osable diret summands of M . As M is stable under the ation of Γ , r ∆  M also and therefore, one an denote b y ∆  M the image of r ∆  M in the real part of a ro ot system of t yp e ∆ . As in [ GLS1 , 3.7℄, there exists a unique w in the W eyl group W of ∆ su h that ∆  M  t α P ∆  | w p α q  0 u where ∆  is the subset of real p ositiv e ro ots of ∆ . Let r w b e the image of w in  W . The subalgebras n p w q (resp. r n p w q ) of n (resp. r n ) are dened as in [GLS1 , 19.3℄. One retains the denitions of the nite dimensional subgroups N p w q and N w (resp. r N p w q and r N w ) of N (resp. r N ) giv en in [GLS1 , 22℄. Then C r N p w qs  U p n p w qq  gr and C r r N p w qs  U p r n p w qq  gr . In the partiular ase where Q is of Dynkin t yp e, w e are in the lassial Lie framew ork. In this ase, let G (resp. r G ) b e the onneted and simply-onneted Lie group asso iated to g (resp. r g ). Then N p w q  N X p w  1 N  w q and N w  N X p B  wB  q ; r N p r w q  r N X p r w  1 r N  r w q and r N r w  r N X p r B  r w r B  q where B  (resp. r B  ) denotes the Borel subgroup of G (resp. r G ) asso iated to b  (resp. r b  ) and N  (resp. r N  ) denotes the unip oten t subgroup of G (resp. r G ) asso iated to n  (resp. r n  ). Here is the analogous of [GLS1 , theorem 3.5℄: Theorem 4.50. The luster algebr a A p B p T qq is a luster algebr a strutur e on C r N p w qs . The luster algebr a r A p B p T qq obtaine d by inversing the  o eients is a luster algebr a strutur e on C r N w s . Pr o of. A ording to [GLS1 , theorem 3.5℄, A p r B p T qq is a luster algebra struture on C r r N p r w qs . It is easy to see that the epimorphism κ : U p r n q  gr { Γ ։ U p n q  gr (see setion 4.1 ) restrits to 58 a b c   @ @ b 1   @ @ a 1 b     a c       @ @ b b 1   @ @ a 1 c   @ @ b 1 c       @ @ b      b 1   @ @ a a 1 c       @ @ b b 1 c b 1   @ @ a 1 c       @ @ b     b 1 a c     b a 1 b 1 c     b     a ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘   ✒   ✒   ✒   ✒     ✒     ✒     ✒ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘    ✒    ✒ ❅ ❅ ❅ ❅ ❅ ❘      ✒       ❅ ❅ ❅ ❅ ❅ ❅       ❅ ❅ ❅ ❅ ❅ ❅       ❅ ❅ ❅ ❅ ❅ ❅       ❅ ❅ ❅ ❅ ❅ ❅ Figure 3. Auslander-Reiten quiv er of Q a morphism U p r n p r w qq  gr { Γ Ñ U p n p w qq  gr . Hene, one an omplete the follo wing omm utativ e diagram : A p B p T qq   / / α      A p r B p T qq{ Γ  / / C r r N p r w qs{ Γ  / / U p r n p r w qq  gr { Γ     / / U p r n q  gr { Γ κ     C r N p w qs  / / U p n p w qq  gr   / / U p n q  gr Moreo v er, using an easy adaptation of prop osition 4.26 , one gets that α is a monomorphism. Using prop osition 4.49 , one onludes that α is an epimorphism, b eause, b y the same metho d as in [GLS1 , prop osition 22.2℄, C r P X s X P  M is the whole algebra C r N p w qs . The ase of C r N w s is handled b y the same metho d.  Example 4.51. In this example, one will denote b y Q the quiv er a b o o c o o / / b 1 / / a 1 endo w ed with the non-trivial ation of Γ  Z { 2 Z . The Auslander-Reiten quiv er of mo d C Q is displa y ed in gure 3. Denote b y M 0 the diret sum of the indeomp osable C Q -mo dules so that C M 0  mo d Λ Q . Let M b e the diret sum of the indeomp osable C Q -mo dules whi h are situated on the righ t of the double line in gure 3. As seen b efore, the group N is of t yp e C 3 . It an b e realized as a subgroup of r N , whi h is seen as a subgroup of the subgroup of GL 6 p C q onsisting of the upp er unitriangular matries. 59 More preisely , N is the subgroup of GL 6 p C q generated b y the one-parameter subgroups         1 t 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 t 0 0 0 0 0 1  Æ Æ Æ Æ Æ Æ          1 0 0 0 0 0 0 1 t 0 0 0 0 0 1 0 0 0 0 0 0 1 t 0 0 0 0 0 1 0 0 0 0 0 0 1  Æ Æ Æ Æ Æ Æ          1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 t 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1  Æ Æ Æ Æ Æ Æ  . As a onsequene, C r N s is a quotien t of C r r N s . In the follo wing table, to a v oid um b ersome indies, w e denote minors of a matrix x P N b y indiating with solid dots the en tries of the orresp onding submatrix of x . Here is the list of the Γ -orbits of the isomorphism lasses of indeomp osable diret summands X of T _ M 0 and the realization of ψ X (see notation 4.10 ) as a minor: Γ -orbit ψ X b   > > c   @ @ b 1   @ @ a 1 b 1   ~ ~ c     b     a             1      1                 1                          1      1     1    1   1  1             c   @ @ b 1   @ @ a 1 c     b     a             1      1     1          1                          1      1     1    1   1  1             c       @ @ b   ; ; ; b 1       @ @ c   = = a 1   ~ ~ b 1 c       @ @ b       = = b 1   ~ ~ a   ? ? c     b             1      1     1       1  1                          1      1     1    1   1  1             c   @ @ b 1 c     b             1      1     1       1  1                          1      1     1    1   1  1             c       @ @ b        ; ; ; b 1   @ @     a   < < < c        = = a 1   ~ ~ b   = = b 1   ~ ~ c             1      1     1    1   1  1             60 Γ -orbit ψ X c       @ @ b   = = b 1   ~ ~ c             1      1     1    1   1  1             c             1      1     1    1   1  1             a   ? ? b   > > c   @ @ b 1   @ @ a 1 a 1   ~ ~ b 1   ~ ~ c     b     a             1                          1                          1      1     1    1   1  1             b       > > a   > > c       @ @ b   ; ; ; b 1       @ @ c   = = a 1   ~ ~ b 1 b 1       @ @ c        = = a 1   ~ ~ b       = = b 1   ~ ~ a   ? ? c     b             1      1             1  1                          1      1     1    1   1  1             The last t w o orbits are those whi h do not app ear in T _ M . A simple omputation sho ws that the elemen t w orresp onding to the ob jet M is w  σ c σ b σ c σ a σ b σ c σ b . As a onsequene, the group N p w q is the subgroup of N onsisting of the matries of the form         1    1     1    1  1 1  Æ Æ Æ Æ Æ Æ  and N w is the sub v ariet y of N onsisting of the matries whi h satisfy             1      1     1    1   1  1                          1      1     1    1   1  1              0 61 and             1      1     1    1   1  1              0 and             1      1     1    1   1  1              0 and             1      1     1    1   1  1              0 . A kno wledgments The author w ould lik e to thank his PhD advisor Bernard Leler for his advies and orre- tions. 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